QUANTITATIVE REASONING-II

PROGRAM: BS EARLY CHILDHOOD CARE AND EDUCATION

(BS-ECCE / B.Ed 4 YEARS)

Code: 8278 Unit 1–9

Early Childhood Education & Elementary Teacher Education Department

FACULTY OF EDUCATION
 QUANTITATIVE REASONING-II
PROGRAM: BS EARLY CHILDHOOD CARE AND EDUCATION
(BS-ECCE / B.Ed 4 YEARS)

Code: 8278 Unit 1–9

Early Childhood Education & Elementary

Teacher Education Department
Faculty of Education
Allama Iqbal Open University
Islamabad

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 DISCLAIMER

The materials for the content development of this course were initially collected and
prepared from several sources. A substantial amount of effort has been made to review and
edit the materials and convert them into this courseware. References and
acknowledgements are given as required. Care has been taken to avoid errors, but errors
are possible. Please let us know of errors or failed links you discover.

Copyright 2024 by AIOU Islamabad

All rights reserved. No part of this publication may be reproduced, stored in a retrieval
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recording scanning or otherwise except as permitted under AIOU copyright ACT.

First Printing .................................................. 2024

Quantity.......................................................... 500
Printer............................................................. Allama Iqbal Open University, Islamabad
Publisher ....................................................... Allama Iqbal Open University, Islamabad

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 COURSE TEAM

Dean: Prof. Dr. Fazal Ur Rahman

Chairman: Dr. Muhammad Athar Hussain

Course Development Coordinator: Dr. Mubeshera Tufail

Writer: Dr. Mubeshera Tufail

Reviewers: 1. Dr. Muhammad Tanveer Afzal

Allama Iqbal Open University, Islamabad

2. Dr. Misbah Javed

Federal College of Education, Islamabad

3. Dr. Muhammad Rafiq

Govt. Higher Secondary School, Khararai

4. Dr. Shahbaz Hamid

Govt. College for Elementary Teachers,
Sharaqpur

Editor: Mr. Fazal Karim

Layout / Typeset by: Muhammad Hameed Zahid

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 TABLE OF CONTENTS

Page #

Foreword ....................................................................................................................... 5

Introduction ................................................................................................................... 6

Objectives of the Course ............................................................................................... 7

Unit–1: Introduction to Critical Thinking ................................................................... 9

Unit–2: Analyzing Arguments .................................................................................... 43

Unit–3: Evaluating Arguments ................................................................................... 63

Unit–4: Sets and Venn Diagrams ................................................................................ 77

Unit–5: Exponential Growth ....................................................................................... 95

Unit–6: Mathematical Models for Understanding Phenomena ................................... 119

Unit–7: Correlation and Causality .............................................................................. 137

Unit–8: Hypothesis Testing......................................................................................... 151

Unit–9: Probability ...................................................................................................... 167

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 FOREWORD

Welcome to this course on quantitative reasoning-II. Quantitative reasoning is not just

about using numbers, it is also about understanding them, interpreting them, and using
them to solve real-world problems. In this course, the fundamental principles of
quantitative reasoning will equip the students with the tools, which are needed to analyze
data, identify patterns and draw meaningful conclusions. In today’s world, where we face
new challenges every day, these skills are more important than ever.

This course strives to present complex ideas in an easily understandable way. It introduces
tools like logical thinking, analyzing arguments, understanding patterns, and calculating
probabilities. These tools will help learners make better decisions in their personal and
professional lives. Every unit in this course is carefully planned to build a strong foundation
for reasoning. From understanding how ideas are connected to evaluating evidence and
solving real-world problems, this course makes learning practical and engaging.

This course would guide you in a step-by-step manner for using quantitative reasoning in
diverse contexts. After completing this course, you would develop the skills to identify the
patterns in the data and draw meaningful insights from the data. The skills developed
through this course would not only be beneficial in your academic and professional journey
but it would be beneficial in the daily life activities

Prof. Dr. Fazal Ur Rahman

Dean, Faculty of Education

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 INTRODUCTION

Quantitative Reasoning-II is a course that helps students to develop important skills for
critical thinking, analyzing information and making better decisions. In today’s world,
these skills are essential for solving problems and understanding complex situations. This
course provides simple tools and techniques to handle daily life challenges with
confidence.

The course begins with Critical Thinking, teaching learners how to solve problems
logically and check the accuracy of information. This continues in the units on Analyzing
Arguments and Evaluating Arguments, where learners explore how to recognize strong
reasoning and avoid common mistakes. Next, the course explains Sets and Venn Diagrams,
showing how to organize information and understand group relationships. In the unit on
Exponential Growth, learners will see how things like populations or investments grow
quickly over time. Mathematical Models for Understanding Phenomena introduces simple
ways to describe real-world problems using equations. Learners will also study Correlation
and Causality to learn the difference between connections and actual cause-and-effect
relationships. In Hypothesis Testing, learners will discover how to test ideas and make
decisions based on evidence. Finally, the course ends with Probability, where learners will
learn how to calculate chances, understand risks, and predict outcomes.

By completing this course, learners will gain problem-solving and reasoning skills. They
will learn to analyze information, evaluate data, and apply logical thinking to everyday
situations. This course is useful for personal growth, career success and effective decision-
making in life.

Dr Mubeshera Tufail
Course Coordinator

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 OBJECTIVES OF THE COURSE

After the successful completion of the course, you would be able to:

1. Define critical thinking and explain its importance in solving problems and making
decisions in daily life.

2. Define problems clearly to apply effective reasoning strategies.

3. Differentiate between inductive and deductive reasoning and explain their uses.

4. Analyze the components of arguments, including premises and conclusions.

5. Evaluate the validity of deductive arguments and the strength of inductive arguments.

6. Use Venn diagrams to show relationships between sets, including overlaps and
disjoint regions.

7. Interpret and analyze numerical data in Venn diagrams to understand how elements
are distributed.

8. Solve real-world problems involving growth and decay using linear and exponential
models.

9. Use the pH scale and logarithmic functions to classify substances and measure
phenomena like sound intensity.

10. Explain correlation and causality, and analyze scatterplots to understand relationships
between variables.

11. Formulate and test hypotheses using statistical techniques to make evidence-based
decisions.

12. Calculate probabilities for different events and apply them to assess risks and solve
problems.

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8
Unit–1

INTODUCTION TO
CRITICAL THINKING

Written by: Dr. Mubeshera Tufail

Reviewed by: Dr. Misbah Javed

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 INTRODUCTION

Critical thinking is an essential skill for daily life. It helps people to make better
decisions, solve problems, and understand information clearly. Instead of just memorizing
facts or accepting things as they are, critical thinking teaches us to carefully analyze,
evaluate, and interpret information in order to form well-thought-out opinions and actions.
It encourages questioning, looking at different perspectives, and finding logical solutions.
This unit focuses on understanding the concept of critical thinking and its importance.
We will learn how to identify and define problems, evaluate extra information, and
understand arguments. The unit will also explain key terms like premises, conclusions, and
arguments, making it easier to apply critical thinking to real-life situations. By the end,
participants will have practical tools to think more clearly and make informed decisions in
their daily lives.
Critical thinking also has a strong link to quantitative reasoning. It helps analyze
numerical data, identify patterns, and make logical decisions based on evidence. Whether
interpreting graphs, solving math problems, or making predictions, critical thinking
ensures that decisions are based on careful reasoning and not assumptions. This makes it a
valuable skill for both everyday life and professional settings.

Learning Outcomes
After the successful completion of the unit, the students would be able to:
1. Understand the concept of critical thinking and its significance in navigating
modern challenges.

2. Identify and define problems effectively to facilitate critical analysis and problem-
solving.

3. Recognize and analyze various types of texts, including descriptions, explanations,

summaries, extraneous material, and arguments.

4. Define key terms related to critical thinking, such as premise, conclusion, and
argument, to enhance comprehension and communication.

5. Apply practical hints and strategies for using critical thinking in everyday life to
make informed decisions and navigate complex information with clarity and
confidence.

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1.1 Introduction to Critical Thinking
Critical thinking is a cognitive activity related to using the mind. Developing the ability to
think critically and analytically involves employing mental processes such as attention,
categorization, selection, and judgment.
Critical thinking is linked to reasoning or our capacity for rational thought. The term
"rational" means using reasons to solve problems. Reasoning begins with ourselves and
involves:
 Having reasons for our beliefs and actions, and being aware of what these reasons
are
 Critically evaluating our own beliefs and actions
 Being able to present to others the reasons for our beliefs and actions
Additionally, it includes:
 Identifying others' reasons and conclusions
 Analyzing how they select, combine, and order reasons to build a line of reasoning
 Evaluating whether their conclusions are well-founded based on good evidence and
logical reasoning

Critical thinking involves developing a range of ancillary skills, including:

 Observation
 Analysis
 Reasoning
 Judgement
 Decision-making
 Persuasion
Critical thinking is a complex process involving a variety of skills and attitudes. It
includes:
 Identifying other people's viewpoints and conclusions
 Evaluating the evidence supporting those viewpoints
 Weighing arguments fairly and opposing positions
 Considering alternative arguments and perspectives
 Reading between the lines to detect hidden assumptions
 Recognizing the techniques that make some arguments more appealing, such as
false logic and persuasive devices
 Reflecting on issues thoughtfully, applying logic and insight
 Drawing conclusions that are valid and justifiable based on solid evidence and
reasonable assumptions
 Presenting a well-reasoned viewpoint in a structured, clear way that convinces
others (Cottrell, 2015)

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1.2 Importance of Critical Thinking
Developing critical thinking skills brings numerous benefits, such as:
 Enhanced attention and observation abilities
 Increased focus on reading and comprehension
 Improved ability to identify key points in a message rather than being distracted
by less important material
 Better ability to respond to the main points in a message
 Greater knowledge of how to apply various analytical skills across different
situations

1.3 Socratic Questions

Critical thinking means carefully reflecting, evaluating, and judging the ideas and
beliefs that shape our own and others' actions. Socratic questioning is a key part of
critical thinking. It encourages deep thinking by asking thought-provoking questions.
There are six types of Socratic questions created by R.W. Paul (Book (1993) Critical
Thinking: How to Prepare Students for a Rapidly Changing World).

S# Type of Socratic Questions Examples

1. Questions for Clarification  Why do you say that?
help ensure that a person  How does this relate to our discussion?
understands the topic clearly  How does this concept apply to real life?
and the conversation remains  Can you explain why this step is
focused. necessary?
2. Questions that probe  What could we assume instead?
assumptions focus on  How can you verify or disapprove that
identifying and challenging the assumption?
ideas or beliefs we take for  Is there any other way to look at it?
granted without questioning  What else could be affecting this
them. These assumptions might phenomenon?
not always be accurate or
applicable. By asking such
questions, we can uncover
hidden biases or explore
alternative perspectives.
3. Questions that probe reasons  What would be an example?
and evidence help us dig  What is....analogous to?
deeper into the reasons behind a  What do you think causes this to
statement or situation. happen...? Why:?
 What would be an example of a healthy
meal?
 What do you think causes the plants not
to grow well? Why do you think that is?
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 4. Questions about Viewpoints  What would be an alternative?
and Perspectives explore  What is another way to look at it?
different ways of thinking or  Would you explain why it is necessary
looking at a situation. or beneficial, and who benefits?
 Why is the best?
 What are the strengths and weaknesses
of...?
 How are...and ...similar?
 What is a counterargument for...?
 "With all the possibilities in mind, from
an industrial/practical standpoint, do
you think it will affect the situation?"
5. Questions that probe  What generalizations can you make?
implications and  What are the consequences of that
consequences assumption?
 What are you implying?
 How does...affect...?
 How does...tie in with what we learned
before?
6. Questions about the question:  What was the point of this question?
 Why do you think I asked this question?
 What does...mean?
 How does...apply to everyday life?
 "Why do you think that this
phenomenon is important?"
(https://websites.umich.edu/~elements/probsolv/strategy/cthinking.htm)

1.4 Identifying the Problem

In the realm of critical thinking, a problem is described as a question or scenario
needing a solution. It implies that encountering a problem requires action or decision-
making to resolve it. Problems presented as questions often lack a single clear answer,
such as, “Why are you voting for candidate X over candidate Y?” or “Why do you
deserve a raise more than Mr. Y?” Situational problems demand critical thinking and
decision-making regarding the best course of action. For example, discovering that a
coworker has inflated company profits on the president's orders raises the dilemma of
whether to report it, risking your career, and to whom to report it.

A common reason for not acknowledging a problem is the reluctance to take action or
responsibility, thinking that ignoring the problem means avoiding responsibility.
However, failing to recognize a problem makes its solution more challenging,
potentially allowing it to grow larger and more complex over time. For instance,

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ordering a new cheque-book before running out of cheques or returning faulty items
before the deadline prevents financial loss.

Upon identifying a problem, determine if it genuinely requires a solution or if it is an

inevitable part of a process. For instance, a new employee making errors on his first
day might not be a problem if such mistakes are expected. Understanding when your
problem-solving skills are necessary is crucial (Starkey, 2004).

1.2.1 Types of Problems

Once you identify a problem, assess its nature based on timeframe and personal
priorities, using severity and importance as criteria.
(i) Severity of Problems
Severe problems have these characteristics:
 Require immediate solutions
 May need help from experts
 Consequences worsen the longer they remain unresolved
For instance, a leaked water tap in your house is a severe problem because water
damage can escalate, requiring immediate professional intervention to prevent
extensive and costly damage to floors, carpets, furniture, and walls. Delays could
necessitate even more expensive repairs or replacements.

Example:
Three problems arise at work simultaneously. In what order do you solve the
following?
1. The printer in your office is down and you need to get the print of an
official report.
2. You need to finish writing the official report to meet a 3:00 P.M. deadline.
3. The report must be dropped off at the courier’s office by 5:00 P.M.

When faced with multiple simultaneous problems at work such as those given
below, prioritizing them based on deadlines and dependencies is essential. Here's the
order in which to solve the given problems:
a) Finish Writing the Official Report (Problem b):
Reasoning: This foundational task must be completed before the other tasks
can be addressed. Without the report being written, there is nothing to print
or drop off. Additionally, it has an immediate deadline (3:00 P.M.), so it must
be prioritized to ensure you meet this critical time constraint.

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 b) Fix the Printer or Arrange for Printing (Problem a):
Reasoning: Once the report is written, the next step is to get it printed. This
step is dependent on the completion of the report, and it needs to be addressed
promptly to avoid delays. If the printer issue cannot be resolved quickly, look
for alternatives such as using a different printer or going to a nearby print
shop.

c) Drop off the Report at FedEx (Problem c):

Reasoning: After the report is written and printed, the final task is to drop it
off at the courier’s office. This task has a later deadline (5:00 P.M.) compared
to the other tasks. By completing the first two tasks efficiently, you ensure
there is enough time left to drop off the report at FedEx before the deadline.

In summary, the order of solving these problems is:

1) Finish Writing the Report (b)
2) Fix the Printer or Arrange for Printing (a)
3) Drop off the Report at FedEx (c)
Tell me what you think about this sequence of tasks.

(ii) Evaluating and Prioritizing Problems

When evaluating and prioritizing problems, it is essential to assess their importance
and urgency to ensure that critical issues are addressed promptly. It ensures that the most
critical and time-sensitive issues are addressed first, preventing more significant problems
from developing while also managing less urgent tasks appropriately. Practicing this
approach will help enhance your critical thinking and problem-solving skills.

Example:
You have the following problems to address today:
a) A leaking roof that might cause water damage to your home.
b) A project deadline at work that needs to be met by the end of the day.
c) An overdue utility bill that must be paid to avoid a service interruption.
d) A scheduled dentist appointment for a routine check-up.
e) A friend’s request to help move furniture to their new apartment.

Step-by-Step Prioritization
Step 1: Identify and List Problems
a) Leaking roof
b) Project deadline at work
c) Overdue utility bill
d) Dentist appointment
e) Friend's request for help

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 Step 2: Assess Each Problem’s Importance and Urgency
a) Leaking Roof
Importance: High (Prevents damage to home)
Urgency: Immediate (Water damage could escalate quickly)
b) Project Deadline at Work
Importance: High (Impacts professional obligations and performance)
Urgency: Immediate (Must be completed by the end of the day)
c) Overdue Utility Bill:
Importance: High (Ensures continued utility services)
Urgency: Immediate (Avoids service interruption)
d) Dentist Appointment:
Importance: Moderate (Important for health but not urgent)
Urgency: Low (Routine check-up can be rescheduled)
e) Friend's Request for Help:
Importance: Low (Social obligation)
Urgency: Low (Can be rescheduled)
Step 2: Rank the Problems in Terms of Priority
a) Leaking Roof: Most important due to the potential for significant damage if not
addressed immediately.
b) Project Deadline at Work: Critical for professional responsibilities and has a
strict deadline.
c) Overdue Utility Bill: Important to ensure the continuity of essential services,
and must be paid to avoid disruption.
d) Dentist Appointment: Important for health but not urgent, can be rescheduled
if necessary.
e) Friend's Request for Help: Least important, a social obligation that can be
postponed.
Conclusion
By evaluating the problems based on their importance and urgency, the ranked order is:
a) Leaking Roof
b) Project Deadline at Work
c) Overdue Utility Bill
d) Dentist Appointment
e) Friend's Request for Help
(iii) Cost of Problem Solving: Prioritizing Repairs
If there are more than one problem involving money to solve and you do not have the
money available to take care of everything all at once, you need to determine what
needs attention first and what can be done later on. When budget constraints limit your
ability to address multiple problems at once, it's essential to prioritize based on
necessity, urgency, and impact on your daily life.

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Example
You need money for two repairs: a car and an air conditioner. You do not have the
money to do both right now. Make a list of the reasons each repair is necessary, and
decide which should be done first.
(i) Car Repair: ______________________________
(ii) Air Conditioner Repair: ____________________
Conclusion: ______________________________

(i) Car Repair

Reasons for Necessity
(a) Safety: A malfunction in the car can affect your car's performance and
potentially lead to more serious issues, compromising safety.
(b) Legal Compliance: A damaged car can lead to noise pollution, and driving
with it may result in fines or penalties.
(c) Daily Commute: If you rely on your car for going to work or other essential
activities, it is crucial to ensure that it is in good working condition to avoid
disruptions.
(ii) Air Conditioner Repair
Reasons for Necessity
(i) Comfort: An air conditioner is important for maintaining a comfortable
indoor environment, especially during hot weather.
(ii) Health: For some individuals, particularly those with respiratory conditions,
a functioning air conditioner can be vital for health.
(iii) Avoid Further Damage: A delay in repairs might cause more significant
issues with the air conditioner, potentially leading to higher repair costs later.

Conclusion
Decision: Prioritize Car Repair
Reasons:
(i) Immediate Safety and Legal Concerns: The car’s issue directly affects safety,
legal compliance, and your ability to use the car for essential activities.
(ii) Daily Necessity: If your car is your primary mode of transportation, ensuring
it is operational takes precedence over the comfort provided by the air
conditioner.
(iii) Potential for Increased Costs: Ignoring the muffler problem could lead to
more severe damage and higher costs down the road, both in terms of repairs
and legal fines.
Therefore, the car repair should be addressed first, while the air conditioner repair
can be scheduled for a later date when funds become available. _________________

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 Activities
 The next time you need to make a TO DO list for yourself, try ranking the
items on your list. You might list them in order of what takes the most or least
time. Or you may list them in order of when they have to be done. You might
have your own order of importance in which to list items or the cost involved.
For practice, try ordering them in each of the different methods listed above.
 Test your skill of problem recognition when watching the news. After you
hear a story, list three problems that will probably occur as a result. (The news
may be related to weather, current affairs or the economy.)

1.5 Defining the Real Problem: A Guide

Identifying the real problem in any situation can sometimes be challenging. Here is a
structured approach to help you define the actual problem effectively:

1) Gather Necessary Information

a) Read Carefully: Examine any feedback or data available. For example, if you
receive a poor grade on an essay, read through it carefully to identify potential
issues.
b) Seek Clarification: If the problem is not apparent, ask for more specific
feedback or information. In the essay example, approach your teacher and
request detailed comments on what needs improvement.
2) Avoid Solving Offshoots Instead of the Core Problem
(a) Identify the Core Issue: Focus on finding the main problem rather than getting
sidetracked by secondary issues. For instance, if your essay has structural
issues, don't just fix minor grammar errors.
(b) Recognize Offshoots: Understand that many smaller problems (offshoots)
may stem from the core issue. Addressing the core issue will often resolve
these offshoots.
3) Do not Be Overwhelmed by Large Problems:
a) Break It Down: Analyze the problem to see if it can be divided into smaller,
more manageable parts. For example, a large project can be broken into tasks
with specific deadlines.
b) Focus on Manageable Parts: Tackle the core issue first, which might
simplify or resolve related problems.

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1.5.1 Steps to Define the Real Problem
1. Identify the Symptoms: List all the apparent issues and symptoms. For example, poor
essay grades, negative feedback, or specific comments from a teacher.
2. Ask Probing Questions: Why is this happening? What is the impact of this issue?
How do these symptoms relate to one another?
3. Gather More Information: If the cause is not clear, seek additional input or feedback.
Speak with others involved (e.g., teachers, and colleagues) to gain different
perspectives.
4. Analyze and Identify the Core Issue: Look for commonalities among the symptoms.
What is the underlying cause that connects various symptoms?
5. Create a Plan to Address the Core Issue: Develop a strategy focused on solving the
main problem. This might involve rewriting the essay with a focus on structure and
clarity if that is the identified issue.
Example: Poor Essay Grade
The step-by-step guide for solving the poor essay grade problem is given below:
Step 1: Identify Symptoms
 Poor grade.
 Vague feedback from the teacher.
Step 2: Ask Probing Questions
 What specific parts of the essay did not meet the standards?
 Is the issue related to structure, content, grammar, or argument
clarity?
Step 3: Gather More Information
 Carefully review the essay.
 Approach the teacher for detailed feedback on specific areas needing
improvement.
Step 4: Analyze and Identify the Core Issue
 If the feedback points to poor argument structure, focus on improving
how arguments are presented and supported.
Step 5: Create a Plan
 Rewrite the essay with an emphasis on clear argumentation and
evidence.
 Seek additional feedback during the rewriting process to ensure
improvement.
Key Reminders:
 Get the information you need: Do not hesitate to ask questions or seek clarification.
 Focus on the main problem: Avoid getting sidetracked by secondary issues.
 Distinguishing between Problems and their Symptoms or Consequences. Avoid
making assumptions based on too little or not very good information. Ask yourself
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  What is really happening?
 What are the underlying causes?
 What are the potential consequences if the problem is not addressed?
 Is there a pattern or trend that indicates a deeper issue?”
 Break down large problems: Tackle manageable parts to make the problem less
overwhelming.
By following these steps, you can effectively identify and address the core
issues in any situation, ensuring that you are solving the real problem rather than
just its symptoms (Starkey, 2004).

1.6 Key Terms in Critical Thinking

 The position is a point of view. For example, artificial intelligence really worries
me.[No reasons are given, so this is simply a position.]
 Agreement is to concur with someone else's point of view. For example, I do not
know much about artificial intelligence, but I agree with you. [No reasons are given,
so this is simply an agreement.] Another example is that “I know a lot about this
subject, and I agree with you. [No reasons are given, so this is simply an agreement.]”.
 Disagreement is to hold a different point of view from someone else. It does not
convince me. I think genetic engineering is really exciting. [No reasons are given, so
this is simply a disagreement.]
 In logic and reasoning, premises are propositions believed to be true and used as the
basis for an argument. They serve as the foundational statements upon which
conclusions are drawn. The validity of an argument largely depends on the truth and
relevance of its premises. If a premise is not well-founded, it is referred to as a false
premise. False premises can undermine the validity of an argument, leading to
erroneous or unsound conclusions. The example of true and false premises are given
below.
Example 1
Premise 1: All mammals are warm-blooded animals. (True Premise)
Premise 2: Whales are mammals. (True Premise)
Conclusion: Therefore, whales are warm-blooded animals.
Example 2
Premise 1: All birds can swim. (False Premise)
Premise 2: Penguins are birds. (True Premise)
Conclusion: Therefore, penguins can swim.
 Proposition: Proposition is a statement that is believed to be true and is presented as
an argument or reason for consideration by the audience. Propositions can vary in
complexity and can cover a wide range of topics or issues. It may be proven true or
false based on evidence and logical analysis. For example,

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 "Climate change is primarily caused by human activities such as burning fossil
fuels."

"Increasing access to education leads to higher rates of economic growth and

development."

"Raising the minimum wage will reduce poverty levels and improve overall
economic well-being."

"Regular exercise is essential for maintaining good physical and mental health."

 Argument is using reasons to support a point of view, with the goal of persuading
known or unknown audiences to agree. An argument may involve disagreement but
extends beyond mere disagreement by being grounded in reasons. An argument
consists of premises and a conclusion. For example,
“Genetic engineering should be curtailed because there has not been sufficient
research into what happens when new varieties are created without natural
predators to hold them in check.”
“The possibilities for improving health and longevity through genetic engineering
offer hope to sufferers of many conditions that currently don't have an effective
cure. We should be pushing ahead to help these people as quickly as we can.”
 Conclusion serves as the endpoint of an argument or piece of reasoning. It is the
culmination of the author's train of thought and should provide a resolution or final
perspective on the issue being discussed. The conclusion should closely align with the
author's main position or thesis statement. It should reflect the author's stance on the
topic and be supported by the evidence and reasoning presented throughout the
argument. In critical thinking, the conclusion is typically a deduction drawn from the
reasons or evidence provided. It is the outcome of logical analysis and evaluation of
the premises. A strong conclusion is one that logically follows from the premises and
effectively supports the author's main argument. It is a crucial element in constructing
sound and persuasive arguments.
 An inductive argument makes a case for a general conclusion from more specific
premises.
Premise 1: Every time I have eaten peanuts, I've had an allergic reaction. (Specific
Observation)
Premise 2: Peanuts contain allergens. (Generalization)
Conclusion: Therefore, eating peanuts will likely cause an allergic reaction.
 A deductive argument makes a case for a specific conclusion from more general
premises.

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 Premise 1: All humans are mortal. (General Principle)
Premise 2: Socrates is a human. (Specific Observation)
Conclusion: Therefore, Socrates is mortal.

Figure 1.1 Patterns of Argument (Harrell & Wetzel, 2015)

1.7 Evaluating an Argument

The criteria for evaluating the respective strengths of inductive and deductive
arguments are given below.
1.7.1 Inductive Argument Evaluation:
(i) Proving Conclusion: Inductive arguments cannot prove their conclusion to
be true with certainty. Instead, they aim to provide evidence that makes the
conclusion probable.
(ii) Strength: Inductive arguments are evaluated based on their strength. A
strong argument presents compelling evidence that supports its conclusion.
(iii) Criteria for Strength: The strength of an inductive argument depends on
how well the evidence supports the conclusion. If the conclusion is well-
supported by the premises and evidence, the argument is considered strong.
If the support is weak or insufficient, the argument is considered weak.

1.7.2 Deductive Argument Evaluation:

(i) Validity: Deductive arguments are evaluated based on validity. A
deductive argument is valid if its conclusion logically follows from its
premises, regardless of the truth of the premises or conclusion.
(ii) Soundness: A deductive argument is sound if it is both valid and all
of its premises are true. Soundness requires not only logical validity
but also the truth of the premises.
These criteria are fundamental in assessing the strength and validity of arguments,
whether they are inductive or deductive in nature. They provide a framework for
evaluating the reliability and persuasiveness of reasoning.

22
 Example of evaluating inductive argument:

Argument: "Every time I have watered the plants, they've grown taller."

Strength: This argument is strong if there is consistent evidence to support the claim.
If the plants have consistently grown taller after being watered, it strengthens the
argument's case.
Weakness: The argument would be weak if there were instances where the plants
didn't grow taller after being watered. Inconsistencies in the evidence weaken the
argument's strength.

Example of evaluating deductive argument:

Argument: "All humans are mortal. Socrates is a human. Therefore, Socrates is

mortal."

Validity: This argument is valid because the conclusion logically follows from the
premises. If it's true that all humans are mortal and Socrates is a human, then it
logically follows that Socrates is mortal.

Soundness: The argument is sound if both premises are true. If it's true that all
humans are mortal and Socrates is indeed a human, then the argument is not only
valid but also sound.

1.8 Recognizing Arguments and Extraneous Materials

Arguments are often embedded within larger pieces of writing and can be
surrounded by various supporting elements to provide context, clarity, and
reinforcement.
1.8.1 Description: Descriptions in writing serve to provide factual and accurate
accounts of how something is done or what something is like. Unlike critical
analysis, they refrain from evaluating outcomes or presenting value
judgments. In reports and academic writing, descriptions aim to convey
information objectively, without bias, and free of any subjective opinions.
While descriptions may delve into detailed examination, their primary
purpose is to offer the audience a thorough impression of the subject matter
being discussed. They do not seek to persuade the audience toward a
particular viewpoint but rather aim to enhance understanding and clarity by
presenting detailed information and factual observations.

23
 Example 1: In the middle of the forest, there's a peaceful spot where sunlight
shines through the tall trees. The trees make a roof of leaves overhead, and
their shadows make patterns on the ground. Sunlight shines through the green
leaves and makes the plants on the ground look colorful. There are moss,
ferns, and pretty flowers everywhere. You can hear birds singing and a stream
nearby. It's quiet and calm, with just the sound of leaves moving and animals
moving around. It feels like time stops in this special part of the forest, and
you can forget your worries and enjoy the peacefulness.
Example 2: The solution was placed in a test-tube and heated to 35 degrees
centigrade. A small amount of yellow vapors were emitted. These were
odorless. Forty milliliters of water were added to the solution, which was then
heated until it began to boil. This time, grey steam was emitted. Water
droplets gathered on the side of the test-tube.
1.8.2 Explanations: Explanations, although resembling arguments in structure,
serve distinct purposes. They may follow a pattern of statements and reasons
leading to a conclusion, akin to arguments, and are often introduced with
signal words familiar in argumentative discourse. However, unlike
arguments, explanations do not aim to persuade the audience to adopt a
specific viewpoint. By elucidating underlying mechanisms or reasons and
illuminating the significance of concepts, explanations provide clarity and
understanding without seeking to shift opinions.
There are two purposes of explanations. (i) Accounting for Why or How:
Explanations primarily aim to provide insight into why or how something
occurs. They seek to clarify the underlying mechanisms or reasons behind a
phenomenon. (ii) Drawing out Meaning: Explanations can also be used to
draw out the meaning of a theory, argument, or other message. They help
illuminate the significance or implications of a concept or idea.
Example 1: "Photosynthesis is a process essential for the survival of plants,
wherein they convert light energy into chemical energy. This occurs primarily
in the chloroplasts of plant cells. During photosynthesis, chlorophyll, the
green pigment in plants, absorbs sunlight. This energy is then used to split
water molecules into oxygen and hydrogen ions. The hydrogen ions are then
combined with carbon dioxide to produce glucose, a simple sugar, and oxygen
is released as a byproduct. This process not only provides plants with the
energy they need to grow and thrive but also plays a crucial role in the Earth's
oxygen cycle, contributing to the oxygen levels in the atmosphere."
Example 2: "The concept of supply and demand is fundamental to
understanding how markets function. In economics, supply refers to the
quantity of a good or service that producers are willing and able to offer for
sale at a given price, while demand represents the quantity of that good or
24
 service that consumers are willing and able to purchase at a given price. When
the price of a product is low, consumers are more willing to buy, resulting in
higher demand. Conversely, when the price is high, consumers may be less
inclined to purchase, leading to lower demand. On the supply side, producers
are more likely to offer higher quantities of a product at higher prices, as it
becomes more profitable for them. The interaction between supply and
demand in the market determines the equilibrium price and quantity, where
the quantity supplied equals the quantity demanded, resulting in a stable
market price."
1.8.3 Summaries: Summaries serve as condensed versions of longer messages or
texts, emphasizing the most crucial points while excluding extraneous details.
They act as reminders of what has been discussed, repeating key ideas to
reinforce understanding. Unlike conclusions, which may introduce new
insights or implications, summaries strictly adhere to previously presented
information. In the context of an argument, the final sentence may serve as a
summary, drawing together the main points to provide closure. However, the
inclusion of new material is uncommon in summaries, as their primary
function is to succinctly encapsulate existing content. Therefore, while
conclusions and summaries may share similarities in their summarization of
information, they serve distinct purposes in communication and are employed
accordingly.
Example 1- Summary of a Scientific Study: "A recent study conducted by
Smith et al. (2023) investigated the effects of sleep deprivation on cognitive
function in young adults. The study found that participants who were deprived
of sleep for 24 hours showed significant impairments in memory, attention,
and decision-making abilities compared to those who had adequate rest.
These findings underscore the importance of sleep for cognitive performance
and highlight the potential risks associated with sleep deprivation, particularly
among young adults. The study suggests that promoting healthy sleep habits
is crucial for maintaining optimal cognitive function."
Example 2 - Summary of an Academic Article: "In their study titled 'The
Impact of Climate Change on Biodiversity in Tropical Rainforests,'
researchers analyzed the effects of global warming on species diversity and
distribution in tropical rainforest ecosystems. Using satellite data and field
observations, the study found that rising temperatures and changes in
precipitation patterns are causing shifts in the geographical ranges of plant
and animal species, leading to alterations in community composition and
ecosystem functioning. The researchers highlight the urgent need for

25
 conservation efforts to mitigate the impacts of climate change on tropical
rainforest biodiversity and ecosystem stability."
Arguments are often embedded within larger pieces of writing and can be surrounded
by various supporting elements to provide context, clarity, and reinforcement.
Introductions serve to orient readers to the topic and provide an overview of the
forthcoming argument. Descriptions offer detailed depictions of phenomena or events
related to the argument, while explanations elucidate the reasoning behind certain claims.
Background information provides context or relevant facts, while summaries recapitulate
key points made within the text. Other extraneous materials, such as examples or
anecdotes, may also be included to support the argument. Together, these elements
contribute to the coherence and persuasiveness of the writing, helping readers grasp the
argument's significance within a broader context.
Example 1
Read the following passage and identify the argument in this passage and various
extraneous materials such as explanation, summary, and background information.
“Climate change is one of the most pressing issues facing our planet today. As global
temperatures rise, the effects on our environment and society become increasingly
evident. Over the past century, the Earth's average temperature has increased by
approximately 1.2 degrees Celsius. This change, largely driven by human activities such
as burning fossil fuels and deforestation, has led to a range of environmental impacts. In
coastal regions, rising sea levels are causing increased flooding and erosion. This not
only affects natural ecosystems but also poses significant risks to human settlements and
infrastructure. The primary reason for sea level rise is the thermal expansion of seawater
as it warms and the melting of ice sheets and glaciers. As temperatures continue to rise,
these processes are expected to accelerate, leading to even higher sea levels. Therefore,
it is crucial for governments around the world to take immediate and decisive action to
mitigate climate change. This includes reducing greenhouse gas emissions, transitioning
to renewable energy sources, and implementing policies to protect vulnerable
communities and ecosystems. For instance, Denmark has made significant strides in
reducing its carbon footprint by investing in wind energy, which now supplies a
substantial portion of the country’s electricity. Similarly, reforestation projects in Brazil
are helping to restore degraded lands and sequester carbon dioxide from the atmosphere.
The evidence clearly shows that climate change poses serious risks to our planet. By
understanding the causes and effects, and by taking proactive measures, we can work
towards a more sustainable future.”

Solution: In this example, the argument that governments need to take action to mitigate
climate change, is surrounded by an introduction, background information, description,
explanation, extraneous materials (examples), and a summary. These elements provide
context, support, and clarity to reinforce the main argument.

26
Introduction: Climate change is one of the most pressing issues facing our planet today.
As global temperatures rise, the effects on our environment and society become
increasingly evident.

Background Information: Over the past century, the Earth's average temperature has
increased by approximately 1.2 degrees Celsius. This change, largely driven by human
activities such as burning fossil fuels and deforestation, has led to a range of
environmental impacts.

Description: In coastal regions, rising sea levels are causing increased flooding and
erosion. This not only affects natural ecosystems but also poses significant risks to
human settlements and infrastructure.

Explanation: The primary reason for sea level rise is the thermal expansion of seawater
as it warms and the melting of ice sheets and glaciers. As temperatures continue to rise,
these processes are expected to accelerate, leading to even higher sea levels.

Argument: Therefore, it is crucial for governments around the world to take immediate
and decisive action to mitigate climate change. This includes reducing greenhouse gas
emissions, transitioning to renewable energy sources, and implementing policies to
protect vulnerable communities and ecosystems.

Extraneous Materials: For instance, Denmark has made significant strides in reducing
its carbon footprint by investing in wind energy, which now supplies a substantial
portion of the country’s electricity. Similarly, reforestation projects in Brazil are helping
to restore degraded lands and sequester carbon dioxide from the atmosphere.

Summary: The evidence clearly shows that climate change poses serious risks to our
planet. By understanding the causes and effects, and by taking proactive measures, we
can work towards a more sustainable future.

Example 2
Read the following passage and identify the argument in this passage and various
extraneous materials such as explanation, summary, and background information.

“The rapid advancement of artificial intelligence (AI) technology is transforming various

sectors, from healthcare to transportation. However, this progression also raises
important ethical and societal questions. Over the past decade, AI has evolved from

27
simple machine learning algorithms to sophisticated systems capable of performing
complex tasks. Companies and governments are increasingly relying on AI to improve
efficiency and make data-driven decisions. In the healthcare sector, AI is being used to
diagnose diseases, recommend treatments, and even predict patient outcomes. These AI
systems analyze vast amounts of medical data to identify patterns that might be missed
by human doctors. The ability of AI to process and analyze large datasets quickly and
accurately is due to advancements in computational power and machine learning
techniques. These technologies enable AI to learn from data and improve its performance
over time. Therefore, it is essential to establish comprehensive ethical guidelines and
regulatory frameworks for the development and deployment of AI. These measures are
necessary to ensure that AI systems are used responsibly and do not infringe on
individual rights or perpetuate biases. For instance, the European Union has proposed
regulations that emphasize transparency, accountability, and fairness in AI.
Additionally, companies like Google and Microsoft have established internal AI ethics
boards to oversee their AI projects and ensure they align with ethical standards. While
AI has the potential to bring significant benefits to society, it also presents ethical
challenges that must be addressed. By implementing robust ethical guidelines and
regulatory frameworks, we can harness the power of AI while safeguarding against its
potential risks.

Solution: In this example, the argument that comprehensive ethical guidelines and
regulatory frameworks for AI are essential is surrounded by an introduction, background
information, description, explanation, extraneous materials (examples), and a summary.
These elements provide context, support, and clarity to reinforce the main argument.

Introduction: The rapid advancement of artificial intelligence (AI) technology is

transforming various sectors, from healthcare to transportation. However, this
progression also raises important ethical and societal questions.

Background Information: Over the past decade, AI has evolved from simple machine
learning algorithms to sophisticated systems capable of performing complex tasks.
Companies and governments are increasingly relying on AI to improve efficiency and
make data-driven decisions.

Description: In the healthcare sector, AI is being used to diagnose diseases, recommend

treatments, and even predict patient outcomes. These AI systems analyze vast amounts
of medical data to identify patterns that might be missed by human doctors.

Explanation: The ability of AI to process and analyze large datasets quickly and
accurately is due to advancements in computational power and machine learning
28
 techniques. These technologies enable AI to learn from data and improve its performance
over time.

Argument: Therefore, it is essential to establish comprehensive ethical guidelines and

regulatory frameworks for the development and deployment of AI. These measures are
necessary to ensure that AI systems are used responsibly and do not infringe on
individual rights or perpetuate biases.

Extraneous Materials: For instance, the European Union has proposed regulations that
emphasize transparency, accountability, and fairness in AI. Additionally, companies like
Google and Microsoft have established internal AI ethics boards to oversee their AI
projects and ensure they align with ethical standards.

Summary: While AI has the potential to bring significant benefits to society, it also
presents ethical challenges that must be addressed. By implementing robust ethical
guidelines and regulatory frameworks, we can harness the power of AI while
safeguarding against its potential risks.

1.9 Critical Thinking in Everyday Life

Critical thinking encompasses a broad set of skills and dispositions beyond mere
isolated abilities. It involves careful reading or listening to understand content deeply,
sharp thinking to evaluate and interpret information, logical analysis to assess
arguments and evidence, and good visualization to imagine scenarios and outcomes.
Additionally, healthy skepticism is essential to question assumptions and seek truth.
Critical thinking cannot be reduced to a simple step-by-step procedure. Instead, it is
mastered through continual practice, experience, and the habit of questioning and
analyzing every argument or decision encountered. A few general guidelines can be
helpful in this process.
1) Hint 1-Pay Close Attention: Language can be intricate, demanding careful effort
to comprehend fully. Always pay close attention when reading or listening to ensure
you understand exactly what is being communicated. Additionally, differentiate
between what was explicitly stated, what was implied, and what needs further
clarification. For example, Let's consider a hypothetical education-related ballot
question:

"Shall there be an amendment to the state constitution to allocate additional funding

for public schools through an increase in property taxes?"

29
 A "yes" vote means supporting the amendment, thus agreeing to allocate
additional funding for public schools through an increase in property taxes. A "no"
vote means opposing the amendment, thus maintaining the current level of funding
for public schools without an increase in property taxes.
This question may spark debate among voters. Some may see a "yes" vote
as a way to improve the quality of education and provide necessary resources to
schools, while others may view it as an unnecessary burden on taxpayers.
Conversely, a "no" vote may be perceived as fiscally responsible by some, while
others may see it as neglecting the needs of students and educators.
2) Hint 2- Uncover Implicit Assumptions: In many arguments, the premises leading
to a conclusion aren't always explicitly stated. Ambiguous terms or hidden
assumptions often underpin these arguments. While the speaker may consider these
assumptions obvious, they might not be apparent to the audience. Thus, what
appears convincing to the speaker might be weak to someone unaware of these
hidden premises. Let us consider an example,
"We shouldn't invest more money in public transportation. It's just a waste of
taxpayer dollars."

The hidden assumption here is that public transportation does not provide
significant benefits to society or that those benefits do not justify the investment of
taxpayer dollars. To evaluate this statement critically, one needs to question
whether public transportation indeed provides no value or if the benefits are being
underestimated. Additionally, it is crucial to examine whether there are alternative
perspectives on the role and importance of public transportation in a community.
3) Hint 3- Uncover the Real Issue: In debates, the true issue might be obscured by
individuals attempting to conceal their true intentions. By carefully analyzing
arguments, it's possible to discern whether the real issue is hidden, even if it's not
explicitly stated. Let us check an example.
"The construction of the new shopping mall must be halted immediately due to
increased traffic congestion in the neighborhood."

The hidden assumption here might be that traffic congestion is the primary concern
driving opposition to the construction of the shopping mall. To uncover the implicit
assumption, one needs to question whether traffic congestion is the only issue at
play. Are there other concerns such as noise pollution, environmental impact, or
changes to the neighborhood's character that are motivating opposition to the
shopping mall? Additionally, it is essential to consider whether alternative solutions
such as implementing traffic management measures, improving public
transportation options, or relocating the shopping mall could address the concerns

30
 while still allowing for economic development. This raises the question of whether
the real issue is about traffic congestion or if there are broader concerns about the
impact of the shopping mall on the community.
4) Hint 4- Consider All Options: In decision-making scenarios where multiple
options are available, it is crucial to thoroughly understand the implications of each
option. Whether choosing between insurance policies, auto loans, or new computer
models, ensuring a comprehensive understanding of how each option would impact
you is key to making informed decisions. For example, let us consider the scenario
of purchasing a new mobile phone. There are three options available, as given
below:
Option A: Purchase the latest model from Brand X.
Option B: Purchase a slightly older model from Brand Y at a discounted price.
Option C: Keep using the current smartphone without upgrading.

To understand all options, let us see the pros and cons of each option.

Option A: Purchasing the latest model from Brand X

Pros: Access to the latest technology, potentially better performance and features.
Cons: Higher cost, and potential compatibility issues with existing accessories or
software.

Option B: Purchasing a slightly older model from Brand Y at a discounted price

Pros: Lower cost, still relatively up-to-date technology.
Cons: May lack some features of the latest model, and the potential for a shorter
lifespan before becoming outdated.

Option C: Keeping the current smartphone without upgrading

Pros: No additional cost, familiarity with the current device.
Cons: Potential for slower performance, missing out on new features and
improvements.

By considering the pros and cons of each option, one can make an informed
decision based on individual preferences, budget, and priorities regarding
smartphone features and technology.
5) Hint 5- Pay Attention to Details: In offers, deals, and contracts, the fine print often
contains critical details that can significantly impact the terms of the agreement.
What initially appears as a favorable deal may prove otherwise when considering
the fine print. Moreover, even the fine print may omit important information.
Therefore, it is essential to exercise critical thinking to identify any missing

31
 information and ensure a comprehensive understanding before making decisions.
Let us consider the scenario of a credit card offer from a bank.

Offer: "Get a new credit card with 0% APR (Annual Percentage Rate) for the first
12 months!"

Fine Print: "0% APR (Annual Percentage Rate) applies only to purchases made
within the first 12 months. Balance transfers and cash advances are subject to a
variable APR of 24.99% after the promotional period. A balance transfer fee of 3%
applies. Late payments may result in a penalty APR of up to 29.99%. Minimum
monthly payments required."

In this example, the initial offer seems appealing with 0% APR (Annual Percentage
Rate) for the first 12 months. However, the fine print reveals that only purchases
qualify for the promotional APR (Annual Percentage Rate), while other
transactions like balance transfers and cash advances accrue interest at a high rate.
Additionally, there are fees associated with balance transfers, and late payments
could result in a penalty APR (Annual Percentage Rate). By paying attention to the
fine print, one can make a more informed decision about whether this credit card
offer aligns with their financial needs and preferences.
6) Hint 6- Consider Alternative Conclusions: When evaluating arguments or
choices, don't settle for the first one presented. Even if an argument supports its
conclusion, there may be other unstated conclusions to consider. Additionally,
inductive arguments, common in real-life scenarios, may appear strong but still lack
conclusive proof for their conclusions. For example, let us consider a scenario.

"The company's profits have increased significantly this quarter, indicating strong
growth in the market. Therefore, it's a good time to invest in the company's stocks."

In this argument, the conclusion is that it is a good time to invest in the company's
stocks.
Supporting Premise is that the company's profits have increased significantly this
quarter, indicating strong market growth. However, alternative conclusions could
also be drawn from the same premise:
Alternative Conclusion 1: The company's profits have increased significantly,
suggesting effective cost-cutting measures rather than genuine market growth.
Alternative Conclusion 2: The company's profits have increased significantly due
to a temporary surge in demand, which may not be sustainable in the long term.
Alternative Conclusion 3: The company's profits have increased significantly, but
market conditions may change unpredictably, posing risks to future profitability.
32
 By considering these alternative conclusions, one can make a more
comprehensive assessment of the situation before making investment decisions.
7) Hint 7- Consider the Big Picture: While details are essential in debates and
arguments, it is crucial not to lose sight of the bigger picture. Focusing too much
on individual details may cause you to overlook the broader context or significance
of the issue at hand. Always step back to ensure you maintain perspective and
understand the overarching themes or implications. Let us consider an example. A
company wants to implement a new policy in a company to increase productivity.
One piece of detail is about employee A. Employee A consistently arrives late to
work. While this detail may be important in discussing employee punctuality and
adherence to company policies, it is essential not to lose sight of the bigger picture:
"The overall productivity of the company is declining, impacting its
competitiveness in the market."
Focusing solely on Employee A's punctuality issue may obscure the larger
problem of declining productivity affecting the company's success. By considering
the broader context, stakeholders can address the root causes of decreased
productivity and develop comprehensive solutions that benefit the organization as
a whole.
1.10 Propositions
Propositions are the building blocks of arguments. It provides the statements upon
which reasoning and conclusions are built. They possess certain characteristics:
(i) Complete Sentence Structure: A proposition must be structured as a
complete sentence i.e., it should have a subject and a predicate, forming
a grammatically coherent statement. For example, "The sun rises in the
east" is a complete sentence and thus qualifies as a proposition.
(ii) Distinct Assertion or Denial: A proposition must make a clear claim
that can be evaluated as either true or false. It presents a statement about
reality, whether it affirms something to be the case or denies it. For
instance, "Dogs are mammals" is a proposition that asserts the truth of
the statement, while "Birds are not vertebrates" is a proposition that
denies the assertion.
By adhering to these criteria, propositions allow for the construction of
coherent arguments and the evaluation of truth claims.
1.1.1 Negation (Negatives)
The negation of a proposition simply states the opposite of the
original proposition. It aims to express the denial or contradiction of
the statement. If we represent a proposition with a letter, such as p,

33
 then its negation is denoted as "not p" or sometimes written as "~
p".
Like propositions, negations also adhere to the structure of a
complete sentence. They present a distinct assertion or denial that
can be evaluated as true or false. For example:

The negation of "Saad is sitting in the chair" is "Saad is not sitting

in the chair."

The negation of "7 + 9 = 2" is "7 + 9 ≠ 2."

Negations play a crucial role in logical arguments, allowing for

the consideration of alternative perspectives and the evaluation of
conflicting claims.
Truth tables are a fundamental tool in logic for systematically
evaluating the truth values of propositions and their negations. They
help to clarify the relationship between propositions and their
corresponding negations. There are two possible values for a
proposition: True (T) and False (F). If a proposition is true (denoted
by T), its negation must be false (denoted by F), and vice versa. We
can represent these facts with a simple truth table—a table that has
a row for each possible set of truth values.

Table 1.1 Truth Table

P Not P
T F This row states that if the
proposition is true (T), then
negation (not p) is false (F)
F T This row states that if the
proposition is false (F), then
negation (not p) is true (T).

Remember:
 Every proposition can be either true (T) or false (F).
 When we negate a proposition, we create another statement that
asserts the opposite claim. It is called negation. It represented
as "not p" or "~ p," always has the opposite truth value of the
original proposition.

34
  A truth table is a table that shows all possible combinations of
truth values for the propositions under consideration, helping to
systematically analyze their relationships.

1.1.2 Logical Connectors (and, or, if…then)

Propositions are often joined together with logical connectors—
words such as “and”, “or” and “if . . . then”. For example, consider
the following two propositions.
(i) Using "and" (Conjunction): When we connect statements
with "and," both statements must be true for the combined
statement to be true. For instance, in the sentence "The test
was hard and I got an A," both conditions need to be fulfilled
for the whole sentence to be accurate. If one of the conditions
is not met (either the test was not hard or you didn't get an
A), then the entire statement becomes false.
(ii) Using "or" (Disjunction): When we link statements with
"or," the combined statement is considered true if at least one
of the individual statements is true. For example, in the
sentence "The test was hard or I got an A," the statement is
true if either part is true. Therefore, if the test was hard but
you did not get an A, or if you got an A but the test was not
hard, the statement remains true.
(iii) "If...then" statements, also known as conditional
propositions: "If...then" statements, also known as
conditional propositions, are crucial in logical reasoning. It
consists of two parts: antecedent and consequent.

Hypothesis (Antecedent): The "If" part of the statement,

denoted as proposition p, represents the condition or
situation assumed to be true. For example, "all politicians are
liars" is the hypothesis.

Conclusion (Consequent): "Then" part of the statement,

denoted as proposition q, represents the outcome or
consequence that is proposed to be true if the hypothesis is
true. For example, "Representative A is a liar" is the
conclusion.

35
 So, the conditional proposition "if all politicians are
liars, then Representative A is a liar" suggests that if the
hypothesis (all politicians being liars) holds true, then the
conclusion (Representative A being a liar) follows as a
consequence. Other examples are given below:
a) If it rains, then the streets will be wet.
b) If you study hard, then you will pass the exam.
c) If the power goes out, then the alarm system will
activate.
However, in real life, the conditionals do not always
appear in “if….then” format. For example, I am not coming
back if I leave.
(iv) Converse: The converse of a conditional statement switches
the positions of the antecedent and the consequent. For
example “If it rains, then the streets will be wet.”, the
converse would be "If the streets are wet, then it is raining."
The converse does not necessarily hold true just because the
original statement is true.
(v) Inverse: The inverse of a conditional statement negates both
the antecedent and the consequent. Following the same
example “If it rains, then the streets will be wet.”, the inverse
would be "If it is not raining, then the streets are not wet."
Like the converse, the inverse does not necessarily hold true
just because the original statement is true.
(vi) Contrapositive: The contrapositive of a conditional
statement involves both switching the positions of the
antecedent and the consequent and negating them. For
example “If it rains, then the streets will be wet.”, the
contrapositive would be "If the streets are not wet, then it is
not raining." Unlike the converse and inverse, the
contrapositive is logically equivalent to the original
conditional statement. If the original statement is true, then
its contrapositive is also true, and vice versa.

36
 Figure 1.1 Conditional, converse, inverse and contrapositive (Briggs, 2015)

Summary
This unit helped us to understand critical thinking, its importance, and its use in daily
life. We learned to recognize and define problems and understand different types of texts,
such as descriptions, explanations, summaries, extra information, and arguments. We also
discussed important terms like premise, conclusion, and argument. Practical tips were
shared to help use critical thinking in everyday situations. By applying the ideas from this
unit, we can improve our ability to think clearly, make better decisions, and understand
information more easily. Since critical thinking skill improves over time, it is important to
keep practicing and reflecting to get better.

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 MULTIPLE-CHOICE QUESTIONS (MCQS)

1. What is the primary goal of critical thinking?

a) To memorize information
b) To analyze and evaluate ideas
c) To follow instructions blindly
d) To avoid asking questions
Answer: b) To analyze and evaluate ideas

2. Which of the following is NOT a type of Socratic question by R.W. Paul?

a) Questions about evidence
b) Questions of clarification
c) Questions about assumptions
d) Questions of power dynamics
Answer: d) Questions of power dynamics

3. What does the term "argument" mean in critical thinking?

a) A heated exchange of words
b) A group of statements leading to a conclusion
c) A statement that cannot be proven
d) Irrelevant information in a discussion
Answer: b) A group of statements leading to a conclusion

4. Which is an example of extraneous material in an argument?

a) Supporting evidence
b) Personal opinion
c) A logical conclusion
d) A proposition
Answer: b) Personal opinion

5. What is the purpose of recognizing propositions in critical thinking?

a) To identify irrelevant information
b) To construct logical arguments
c) To memorize definitions
d) To write persuasive essays
Answer: b) To construct logical arguments
38
6. What is a conclusion in an argument?
a) A random statement
b) A premise
c) A statement derived from premises
d) A reason for a discussion
Answer: c) A statement derived from premises

7. Which of the following is an example of critical thinking in everyday life?

a) Checking the credibility of a news article
b) Memorizing all facts in a book
c) Following instructions
d) Sticking to your own beliefs
Answer: a) Checking the credibility of a news article

8. What is the role of logic in critical thinking?

a) To avoid asking challenging questions
b) To memorize information without understanding it
c) To make decisions without evidence or analysis
d) To guide reasoning and evaluate arguments
Answer: d) To guide reasoning and evaluate arguments

9. What makes an argument valid?

a) Logical connections between premises and conclusion
b) Emotional appeal and persuasive language alone
c) The repetition of ideas without new evidence
d) The inclusion of opinions without supporting facts
Answer: a) Logical connections between premises and conclusion

10. Which of the following reflects critical thinking in everyday life?

a) Believing claims without checking their credibility
b) Following trends without questioning their validity
c) Analyzing reviews before purchasing a product
d) Accepting all advertisements as factual and accurate
Answer: c) Analyzing reviews before purchasing a product

39
Short-Answer Questions
1. Define critical thinking in your own words.
2. List two reasons for the importance of critical thinking in daily life.
3. Explain the concept of an argument in the context of critical thinking.
4. What are propositions, and why are they essential for logical reasoning?
5. Provide an example of extraneous material in an argument.
Subjective Questions
1. Explain the importance of Socratic questioning in developing critical thinking skills.
2. Discuss the role of logic and reasoning in constructing sound arguments.
3. How can critical thinking help in evaluating news or social media claims? Provide
examples.
4. Explain the relevance of critical thinking in the problem-solving process.
5. Compare and contrast arguments with extraneous materials. Why is it crucial to
differentiate between arguments and other extraneous materials?
6. How do you plan to apply the knowledge and skills gained from this unit to your
future academic or professional endeavors?
7. Based on your understanding of this unit, what steps do you plan to take to improve
your critical thinking skills in the future?

40
Activities
1) Select a recent news article on a topic of interest. Answer the following:
i. What is the main argument of the article?
ii. Are the facts presented supported by evidence?
iii. Are there any assumptions or biases?
iv. How reliable is the source?
2) Choose a simple topic (e.g., "Online education is better than traditional education").
Write:
i. A clear proposition.
ii. At least three supporting premises.
iii. A logical conclusion.
3) Choose a controversial statement (e.g., "Social media does more harm than good").
Ask about it the following types of Socratic questions:
i. Clarification: "What is meant by harm?"
ii. Assumptions: "What assumptions are being made here?"
iii. Evidence: "What evidence supports this claim?"
iv. Alternatives: "What could be another perspective on it?"
v. Implications: "What might happen if this statement is true?"
4) Read the following argument, identify the extraneous material and explain why it is
unnecessary.
"Eating fruits daily improves health. Studies show that fruits contain essential
vitamins. Recently, there has been a rise in organic farming. Including fruits in
your diet reduces the risk of certain diseases."
5) Reflect on one of your recent decisions (e.g., purchasing a product, or solving a
problem) and analyze:
i. What options did you consider?
ii. How did you evaluate these options?
iii. Was your decision logical and evidence-based?
iv. Could you have made a better decision?
6) Watch or read an advertisement for a product and answer the following questions:
i. What claims are made in the advertisement?
ii. Are these claims supported by evidence?
iii. Are there any logical fallacies (e.g., appeals to emotion, false cause)?
iv. Would you trust this product based on the ad? Why or why not?
7) Solve a simple logic puzzle (e.g., "If all A are B and all B are C, are all A also
C?"). Explain your answer and the reasoning process for it.
8) Post a question on the AIOU LMS/MS Teams/LMS discussion forum (e.g., "Is
technology always beneficial for education?"). Invite responses from at least two
colleagues using the Socratic questioning technique. Reflect on how the discussion
deepened your understanding of the topic.

41
 REFERENCES

Briggs, B.(2015). Using and understanding Mathematics: A quantitative reasoning

approach. USA: Pearson Education Inc.

Cottrell, S. (2015). Critical Thinking Skills. New York: Palgrave Macmillan Ltd.

Harrell, M. & Wetzel, D. (2015). Using Argument Diagramming to Teach Critical

Thinking in a First-Year Writing Course. The Palgrave Handbook of Critical

Thinking in Higher Education. DOI: 10.1057/9781137378057_14

Starkey, L. (2004). Critical thinking skills success in 20 minutes a day. New York:
Learning Express, LLC.

The Six Types of Socratic Questions. (n.d.). Retrieved from

https://websites.umich.edu/~elements/probsolv/strategy/cthinking.htm

42
Unit–2

ANALYZING ARGUMENTS

Written by: Dr. Mubeshera Tufail

Reviewed by: Dr. Misbah Javed

43
 INTRODUCTION

Critical thinking is an important skill that helps us make informed decisions, solve
problems, and evaluate information effectively. It involves looking at facts, questioning
assumptions, and carefully analyzing arguments before making conclusions. To think
critically, we need different methods of reasoning, which guide us in understanding
complex issues and drawing logical conclusions.
One key way to reason is through deductive reasoning. This method starts with general
principles or rules and applies them to specific situations to reach a certain conclusion.
Deductive reasoning is a powerful tool for verifying information and ensuring that
decisions are based on facts. The other method, inductive reasoning, works in the opposite
direction. It involves looking at specific examples or observations and making general
conclusions from them. Inductive reasoning helps us form hypotheses and develop theories
based on evidence.
Both inductive and deductive reasoning are connected to quantitative reasoning, as they
both rely on analyzing data, recognizing patterns, and drawing conclusions based on
evidence. By mastering these reasoning techniques, we can enhance our critical thinking
skills, allowing us to make better decisions and solve problems more effectively in
everyday life and in academic or professional settings.
Difference between Fact and Opinion
Facts are objective statements that can be verified through research or observation. They
represent truths that are universally accepted or can be confirmed to be true. For instance,
"Saturn is one of the nine planets in the solar system" is a fact because it can be verified
by scientific evidence. When you read such a statement in a document, you can confirm
this information from some authentic source. If the statement is always true then it is a
fact.

On the other hand, opinions are subjective statements based on personal beliefs or
preferences. They reflect individual viewpoints and are not necessarily true for everyone.
For example, "Saturn is the most beautiful planet in the solar system" is an opinion
because beauty is subjective and varies from person to person. When you read such a
statement in a document, check whether the statement is true for everyone. If the answer
is no then it is an opinion.

Distinguishing between facts and opinions is important in critical thinking and

communication, as it helps to differentiate between objective reality and personal
perspectives.

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 Learning Outcomes
After the successful completion of this unit, the students will be able to:
1. Define inductive and deductive reasoning.
2. Differentiate between inductive and deductive arguments based on their structure
and purpose.
3. Analyze the components of inductive and deductive arguments, including
premises and conclusions.
4. Evaluate the validity and soundness of deductive arguments.
5. Assess the strength and reliability of inductive arguments.
6. Enhance critical thinking skills by evaluating the strength of various arguments.
7. Make informed decisions by applying principles of inductive and deductive
reasoning.
8. Recognize the use of inductive and deductive reasoning in everyday life, such as
in scientific inquiry, legal reasoning, and personal decision-making.

2.1 Deductive Argument

Deductive reasoning involves forming an argument based on two premises, assuming
they are true, thereby concluding that the argument's conclusion must also be true. We
encounter deductive arguments frequently, some effective and others not. For instance, in
advertisements, you may encounter claims like "Using Brand X detergent won't clean your
clothes, but our detergent will." Similarly, politicians may argue, "High taxes lead to
unemployment; therefore, tax cuts are preferable as they create jobs." Even in everyday
situations, parents might say, "No supper, no dessert." Understanding the mechanics of
these arguments, both their strengths and weaknesses, serves two purposes. First, it enables
one to construct robust arguments using deductive reasoning, facilitating clearer
communication of ideas. Second, it empowers individuals to discern weak arguments,
protecting against being swayed by flawed reasoning. Simultaneously, it allows for
recognizing compelling arguments that warrant consideration.
Definition of Deductive Argument
Deduction involves reasoning from two broad premises or established facts to arrive at
a specific conclusion. This process comprises three components:

A. Major premise
B. Minor premise
C. Conclusion

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 For instance, considering the known facts that A) dogs typically have four legs, and
B) Fido is identified as a dog, we can logically infer that C) Fido has four legs. This
example illustrates that a deductive argument seems sound when its premises are true,
and the conclusion logically stems from these premises.

Examples of Weak and Strong Deductive Arguments

Weak Deductive Argument
Premise 1: All humans have wings.
Premise 2: John is a human.
Conclusion: Therefore, John has wings.

This argument is weak because the first premise is clearly false. It is not a general truth
that all humans have wings, so the conclusion does not logically follow from the
premises.

Strong Deductive Argument

Premise 1: All mammals are warm-blooded.
Premise 2: Dogs are mammals.
Conclusion: Therefore, dogs are warm-blooded.

This argument is strong because both premises are true and the conclusion logically
follows from them. It adheres to the principles of deductive reasoning, where the
conclusion necessarily follows from the premises.

2.1.1 Characteristics of a Deductive Argument:

Three characteristics of the deductive argument are given below:
1. It comprises two premises that offer assurance regarding the truth of the
conclusion by providing such robust support that, if the premises hold true, the
conclusion could not possibly be false.
2. It is categorized as either valid or invalid; when the premises are accurate and
the conclusion logically follows, the argument is considered valid. If one or
both premises are flawed, the argument becomes invalid.
3. It relies on rules, laws, principles, or generalizations, as opposed to inductive
arguments, which derive their major premises from observations or
experiences.

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 Examples
1. a. There are 25 CDs on the top shelf of my bookcase and 14 on the lower shelf.
There are no other CDs in my bookcase. Therefore, there are 39 CDs in my
bookcase.

This argument presents two premises (the number of CDs on each shelf) and
draws a conclusion based on those premises. It follows the structure of a
deductive argument, where the conclusion logically follows from the premises.

2. Nadia is either in Rawalpindi or Islamabad. If Nadia is in Rawalpindi, then Nadia

is in Punjab. If Topeka is in Islamabad, then Nadia is in Islamabad Capital
Territory. Therefore, Topeka is in Rawalpindi.

This argument has three premises and the conclusion does not follow from them.

3. No one got an A on yesterday’s class test. Ali was not in school yesterday. Jimmy
will get the higher score on the test today, and get an A.

This argument has the conclusion that do not follow the premises.

All human beings are in favor of world peace. Terrorists do not care about world
peace. Terrorists bring about destruction.

This argument has the conclusion that do not follow the premises.

2.1.2 Components of Deductive Argument

In a deductive argument, there are two key components: the premises and the
conclusion, as given below:
(i) Premises
Premises are the statements or propositions that serve as the foundation for
the argument. Premises are accepted as true or assumed to be true for the
sake of the argument. They provide the evidence or support upon which the
conclusion is based.
The validity of a deductive conclusion hinges on the accuracy of its
premises. If any of the premises are proven false, the conclusion is deemed
invalid since it must logically follow from the premises. Thus, the premises
must consist of truthful facts, rules, principles, or generalizations. Even a

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 single word alteration can transform a premise from fact to fiction, as seen
with terms like "all" and "every." For example,
 All dogs have brown fur.
 Spot is a dog.
 Spot has brown fur.
While the conclusion seems logical, the truth is that only some dogs
have brown fur, not all. This makes the first premise untrue, thus
invalidating the conclusion.
Major Premise and Minor Premise
The major premise serves as a statement of general truth, focusing on
categories rather than individual instances. It establishes a relationship
between two terms: antecedent and consequent. For example,
 All women were once girls.
 Athletes are in good shape.
 Professors hold advanced degrees.

The antecedent refers to the subject of the major premise, such as

women, athletes, or professors. The consequent is the verb phrase that
describes the characteristic or trait associated with the antecedent, like
"were once girls," "are in good shape," or "hold advanced degrees."

The minor premise deals with a specific instance of the major premise.
It either confirms or denies the major premise. Based on the above-
mentioned examples of major premises, minor premises are given below:
 My mother is a woman.
 Tiger Woods is an athlete.
 Dr. Shiu is a professor.
When affirming, the minor premise aligns with the subject or antecedent
of the major premise. When denying, the minor premise contradicts the
consequent of the major premise. For example,
Affirming
Major Premise: Children like top 40 music.
Minor Premise: Charles is a child.
(Here, the minor premise confirms the major premise by stating something
consistent with it.)
Denying:
Major Premise: Children like top 40 music.
Minor Premise: Charles does not like top 40 music.

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 (In this case, the minor premise contradicts the major premise by asserting
something different from its consequent.)
(ii) Conclusion
Conclusion is the statement or proposition that logically follows from
the premises. It is the ultimate claim or inference drawn from the premises.
The conclusion is the main point or the assertion that the argument seeks to
establish.
In deductive arguments, the truth of the conclusion is guaranteed by the
truth of the premises. When an argument is valid, it means that the truth of
the conclusion is inherently contained within the truth of the premises. For
example:
Premise 1: Banks make money by charging interest.
Premise 2: My bank charges me interest.
Conclusion: My bank makes money.

The conclusion logically follows from both premises without

introducing any new information or making unwarranted assumptions. It
directly stems from the premises, making it a valid conclusion. This
exemplifies the essence of deductive reasoning, where the conclusion is a
natural consequence of the premises.
Example 1
Premise 1: Author ‘A’ wrote some great books.
Premise 2: Author ‘A’ wrote Critical Thinking Skills.
Conclusion: Critical Thinking Skills is a great book.

The conclusion goes beyond the truth of the premises because the
fact that Author ‘A’ wrote some great books, does not necessarily mean
that every book s/he wrote is great. Therefore, while the premises might
support the possibility that Critical Thinking Skills is a great book, it does
not provide a guarantee, making the conclusion extend beyond the scope
of the premises. It makes the argument invalid in a deductive context.

Example 2
The price of every daily newspaper is going up next week. The News is a
daily newspaper. Therefore, The News’ price will double next week.

The conclusion should be: ‘Therefore, the price of The News will go
up next week.’ The deductive argument does not say the price will be
double.

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 Identifying and understanding these parts is crucial for recognizing
and evaluating deductive arguments. By analyzing the premises and the
conclusion, one can assess the validity of the argument and determine
whether the conclusion logically follows from the premises.
2.1.3 Forms of Deductive Argument
There are two common ways in which deductive arguments are expressed:
syllogism and conditionals.
(1) Syllogisms
Syllogisms consist of two premises and a conclusion.
 The major premise describes one class or group in terms of another
class or group (All A are not B). Example: All vegetarians do not eat
meat.
 Minor premise places a third class or group within the first or not within
the second (C is not B). Example: Shahid does not eat meat.
 The conclusion affirms the relationship between the third class and the
second class (C is A). Example: Shahid is a vegetarian.
Examples of Syllogisms:
a) All birds have feathers. A robin is a bird. A robin has feathers.
b) All squares are rectangles. All rectangles have four sides. All
squares have four sides.
c) No cars can fly. A Toyota Camry is a car. A Toyota Camry cannot
fly.
(2) Conditionals
Conditionals express the same reasoning but in a different format.
 The major premise establishes a conditional relationship between two
events or states (If A, then B). Example: If Ali stays after class to speak
with his professor, he will miss the bus.
 The minor premise either confirms or denies the occurrence of the first
event or state (A). Example: Jason did not stay after class to speak with
his professor.
 The conclusion follows based on the confirmation or denial of the first
event or state (B). Example: Jason did not miss the bus.
Examples of Conditionals:
a. If a person attends the nearby Gym, s/he will lose weight (If A, then
B). You attend the nearby Gym (A). You lose weight (B).
b. If we do not negotiate with the other team, they will defeat us (If
not A, then B). We negotiated (A). They did not defeat us (not B).
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 c. If you do not play games, you will get sick. You played games. So,
you did not get sick.
Example to differentiate between Syllogisms and Conditionals

Sana says that all her test scores are good, so the grades for her courses
should be good, too.

Syllogism: All good test scores mean good course grades. Sana’s test
scores are all good. Sana gets good course grades.
Conditional: If you get good test scores, then you get good course grades.
Sana gets good test scores. Therefore, she gets good course grades.

2.1.1 Valid or Invalid Deductive Argument

In deductive arguments, the truth of the conclusion is guaranteed by the truth of
the premises. When an argument is valid, it means that the truth of the conclusion is
inherently contained within the truth of the premises. For example:

Premise 1: Banks make money by charging interest.

Premise 2: My bank charges me interest.
Conclusion: My bank makes money.

The conclusion logically follows from both premises without

introducing any new information or making unwarranted assumptions. It
directly stems from the premises, making it a valid conclusion. This
exemplifies the essence of deductive reasoning, where the conclusion is a
natural consequence of the premises.
2.1.2 Reasons for Validity or Invalidity of Deductive Argument
A deductive argument is considered invalid for one of two possible
reasons:
a) Invalid Premises: One or both of the premises are not true.
Example of an invalid premise where one of the premises is not true:

Premise 1: All cats are reptiles.

Premise 2: Whiskers is a cat.
Conclusion: Whiskers is a reptile.

In this case, Premise 1 is not true because cats are mammals, not
reptiles. Therefore, the conclusion is invalid because it is based on a
false premise.

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 b) Invalid Conclusion: The conclusion does not logically follow from
the premises, even if the premises are true. Example of an invalid
conclusion where the premises are true, but the conclusion does not
logically follow:

Premise 1: All birds have wings.

Premise 2: A penguin is a bird.
Conclusion: A penguin can fly.

Here, both premises are true. However, the conclusion is invalid

because, while all birds have wings, not all birds can fly. Penguins are
an example of birds that cannot fly. Therefore, the conclusion does not
logically follow from the premises.
By examining the validity of the premises and the logical
connection between the premises and the conclusion, we can determine
whether a deductive argument is valid or invalid.
2.2 Inductive Argument
Inductive reasoning involves drawing general conclusions from specific
observations or facts. Unlike deductive reasoning, which guarantees the truth of the
conclusion if the premises are true, inductive reasoning only suggests that the
conclusion is probable. The structure of an inductive argument typically looks like this:

Premise 1: A is true.
Premise 2: B is true.
Conclusion: Therefore, C is probably true.

Example: Cell theory i.e., one of the basic theories of modern biology, cell theory, is a
product of inductive reasoning
Premise 1: Every organism that has been observed is made up of cells.
Conclusion: Therefore, it is most likely that all living things are made up of cells.

Determining the Probability of the Conclusion based on premises

To determine or measure the probability of the conclusion based on premises, we rely
on:
i. Past Experience [For example: "For the past three weeks, my colleague
has shown up a half hour late for work. Today, she will probably be late
too."]

52
 ii. Common Sense [For example: "They need five people on the team. I’m
one of the strongest of the seven players at the tryouts. It’s likely that I
will be picked for the team."]
Examples of Strong Inductive Arguments
In the context of inductive reasoning, "valid" typically means "strong" rather than
absolutely certain, as inductive arguments deal with probability rather than certainty.

Example 1
Premise 1: In a clinical trial, 95 out of 100 patients who took Drug X experienced a
significant reduction in symptoms.
Conclusion: Therefore, it is likely that Drug X is effective in reducing symptoms for
most patients who take it.

It is a strong inductive argument because the conclusion is based on a significant

amount of observed data from the clinical trial.

Example 2
Premise 1: In the past 20 years, the trees in the park have started changing color in
mid-September.
Conclusion: Therefore, the trees in the park will likely start changing color in mid-
September this year.

This argument is strong because it is based on consistent, long-term observations that

suggest a pattern.

Example 3
Premise 1: The last three smartphones I bought from Brand Y have been high quality
and lasted for several years.
Conclusion: Therefore, the next smartphone I buy from Brand Y will likely be high
quality and last for several years.

This argument is strong because it is based on repeated personal experience with a

specific brand, suggesting a pattern of quality.

2.2.1 Forms of Inductive Arguments

There are two forms of inductive arguments, as given below:
i. Comparison Arguments: Comparison arguments compare one
thing, event, or idea to another to see if they are similar. For example

53
 "This smartphone model has excellent reviews and long battery life,
similar to the previous model from the same brand. Therefore, this new
model will probably have long battery life as well.

A comparison argument involves comparing one event, idea, or

thing with another to establish that they are similar enough to make a
generalization or inference about them. The most important point to
note about this type of argument is that the two events being compared
must be similar. However, the strength of these arguments depends on
the degree of similarity between the compared entities. If the
similarities are superficial or irrelevant, the argument may be weak.

Examples of Comparison Arguments

Example 1: Education Systems

Premise 1: Finland's education system emphasizes student well-being and
consistently ranks high in international assessments.
Premise 2: Canada has similar social and economic conditions to Finland.
Conclusion: Therefore, adopting Finland’s educational practices in Canada
could likely improve its educational outcomes.
Example 2: Business Strategies
Premise 1: Company A implemented a four-day workweek, leading to higher
employee satisfaction and productivity.
Premise 2: Company B operates in the same industry and has a similar corporate
culture to Company A.
Conclusion: Therefore, Company B could also benefit from implementing a
four-day workweek.
Example 3: Public Health Policies
Premise 1: Country X reduced smoking rates by imposing higher taxes on
cigarettes and banning smoking in public places.
Premise 2: Country Y has a similar smoking rate and demographic profile to
Country X.
Conclusion: Therefore, Country Y could reduce its smoking rates by adopting
similar policies.

ii. Causal Arguments: Causal arguments try to establish a causal

relationship between two events or phenomena based on observed
patterns or correlations. For example

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 "Every time it rains, the football field floods. It’s raining today, so the
football field will probably flood."
Real-life situations can get complicated. Our lives and the world
around us are affected by thousands of details, making it challenging
to identify one key difference. However, if there is a strong likelihood
of causation and no other obvious causes, you can make a convincing
causal argument. To do so, you need to ensure the following:
 Effect Must Occur After the Cause: It seems like common
sense, but many arguments mistakenly place the effect before
the cause. For example, you are blamed for a computer problem
at work. However, you did not use the computer until after the
problem was detected. The argument against you is therefore
invalid.
 More Than Just a Strong Correlation is Needed to Prove
Causation: Coincidence can often explain what might initially
appear to be a cause-and-effect relationship. For example,
every time you wear your blue sweater, your team wins the
game. Can you conclude that because you always wear the
sweater, your team always wins? The answer is no, because
there is no causation. Nothing about wearing the sweater could
have caused the team’s win.
Examples of Causal Arguments
Example 1: Effect of Exercise on Mental Health
Observation: Individuals who engage in regular exercise often report
improved mental well-being.
Observation: People with sedentary lifestyles are more likely to
experience symptoms of depression and anxiety.
Conclusion: Regular exercise may have a positive impact on mental
health.

Example 2: Influence of Parental Involvement on Academic

Achievement
Observation: Students whose parents actively participate in their
education tend to perform better academically.
Observation: Children from households where parental involvement is
low often struggle academically.
Conclusion: Parental involvement likely contributes to students'
academic success.

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 Example 3: Relationship Between Smoking and Respiratory
Illnesses
Observation: Individuals who smoke tobacco have a higher incidence
of respiratory illnesses such as bronchitis and emphysema.
Observation: Non-smokers generally have lower rates of respiratory
illnesses compared to smokers.
Conclusion: Smoking tobacco is likely a contributing factor to the
development of respiratory illnesses.

Example 4: Impact of Sleep on Cognitive Function

Observation: People who consistently get adequate sleep demonstrate
better cognitive function and memory retention.
Observation: Individuals who experience chronic sleep deprivation
often exhibit cognitive impairment and difficulty concentrating.
Conclusion: Sufficient sleep likely plays a crucial role in maintaining
cognitive function.

Example 5: Influence of Diet on Heart Health

Observation: Individuals who consume a diet high in fruits, vegetables,
and lean proteins have lower rates of cardiovascular diseases.
Observation: People with diets high in processed foods and saturated
fats are more prone to heart-related issues.
Conclusion: Dietary habits likely influence heart health outcomes.

2.3 Key Differences: Inductive and Deductive Arguments

Inductive Arguments Deductive Arguments
A conclusion is formed by generalizing A specific conclusion is deduced from a
from a set of more specific premises. set of more general (or equally general)
premises.
An inductive argument can be analyzed A deductive argument can be analyzed in
only in terms of its strength. Evaluating terms of its validity and soundness:
strength involves personal judgment  It is valid if its conclusion
about how well the premises support the follows necessarily from its
conclusion. premises.
 It is sound if it is valid and its
premises are true.

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 An inductive argument cannot prove its Validity concerns only logical structure a
conclusion true. At best, a strong deductive argument can be valid even
inductive argument shows that its when its conclusion is blatantly false.
conclusion probably is true.

2.4 Application of Critical Thinking in Everyday Life

Inductive and deductive reasoning are essential tools used in various aspects of
daily life, ranging from problem-solving to decision-making. Some common
applications of Inductive and deductive reasoning are explained below.
2.4.1 Applications of Inductive Reasoning
(i) Weather Forecasting: Meteorologists use historical weather data and
patterns to make predictions about future weather conditions. By
observing trends in temperature, humidity, and atmospheric pressure,
they can forecast upcoming weather events.
(ii) Healthcare Diagnosis: Doctors often use inductive reasoning when
diagnosing illnesses. They collect symptoms and medical history from
patients, look for patterns or similarities with known diseases, and
make a diagnosis based on these observations.
(iii) Market Research: Businesses use inductive reasoning to analyze
consumer behavior and market trends. By studying past sales data and
consumer preferences, they can predict future market demand and
tailor their products or services accordingly.
(iv) Educational Assessment: Teachers use inductive reasoning to assess
students' learning progress. By observing students' performance on
assignments and tests over time, they can identify patterns of strengths
and weaknesses and adjust their teaching strategies accordingly.
(v) Criminal Profiling: Law enforcement agencies use inductive
reasoning in criminal investigations. By analyzing patterns in crime
scenes and suspects' behavior, they can develop profiles of potential
offenders and narrow down their search for suspects.
2.4.2 Applications of Deductive Reasoning
(i) Legal Decision-Making: Lawyers and judges use deductive reasoning
to apply legal principles to specific cases. They start with general legal
rules or statutes and then apply them to the facts of a case to reach a
logical conclusion or verdict.
(ii) Mathematical Problem-Solving: Mathematicians use deductive
reasoning to prove theorems and solve mathematical problems. They
start with axioms and logical rules and then apply them systematically
to derive new mathematical truths.
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 (iii) Computer Programming: Programmers use deductive reasoning to
write code and develop algorithms. They start with logical rules and
programming syntax and then apply them to solve specific
computational problems or tasks.
(iv) Critical Thinking: Individuals use deductive reasoning in everyday
problem-solving and decision-making. By applying logical rules and
principles, they can evaluate arguments, identify fallacies, and draw
valid conclusions based on evidence and reasoning.
(v) Scientific Research: Scientists use deductive reasoning to formulate
hypotheses and test them through experiments. They start with a
hypothesis based on existing theories or observations and then design
experiments to gather data and confirm or refute the hypothesis.
In summary, inductive and deductive reasoning are integral parts of
everyday life, used in various fields such as healthcare, business, law,
education, and science. Both forms of reasoning help individuals make sense
of complex information, make informed decisions, and solve problems
effectively.
Summary
This unit explains the importance of inductive and deductive reasoning in critical
thinking and decision-making. Both methods help us understand how to reason and solve
problems in everyday life and different fields of study. Inductive reasoning looks at
patterns and evidence to make predictions and draw conclusions. It is useful in areas like
science, market research, and healthcare. For example, by observing trends, we can make
informed guesses about future events or create hypotheses to test. Deductive reasoning
focuses on using general rules or principles to solve specific problems. It is essential in
fields like mathematics, law, and computer science. By following logical steps, we can
reach clear and accurate conclusions.

58
Multiple-Choice Questions
1) What is a defining characteristic of a deductive argument?
a) It is based on probability.
b) It provides evidence for generalizations.
c) It guarantees the truth of the conclusion if the premises are true.
d) It relies on statistical data.
Answer: c) It guarantees the truth of the conclusion if the premises are true.
2) Which of the following is NOT a component of a deductive argument?
a) Premises
b) Conclusion
c) Assumption
d) Probability
Answer: d) Probability
3) Which of these best describes an invalid deductive argument?
a) The premises are true, and the conclusion is true.
b) The premises are true, but the conclusion does not logically follow.
c) The premises and conclusion are false.
d) The argument is based on probability.
Answer: b) The premises are true, but the conclusion does not logically follow.
4) What is the primary goal of an inductive argument?
a) To guarantee the conclusion.
b) To provide irrelevant premises.
c) To present evidence for the conclusion.
d) To avoid generalizations.
Answer: c) To present evidence for the conclusion.
5) Which of these is an example of inductive reasoning?
a) All humans are mortal. Socrates is a human. Therefore, Socrates is mortal.
b) The sun has risen every day in the morning. Therefore, it will rise tomorrow.
c) A square has four equal sides. A shape is a square. Therefore, it has four sides.
d) Most students in this school enjoy sports. Maria is a student here. Therefore, Maria
likely enjoys sports.
Answer: b) The sun has risen every day in the morning. Therefore, it will rise
tomorrow.
6) How can one distinguish between inductive and deductive arguments?
a) Deductive arguments rely on evidence; inductive arguments rely on facts.

59
 b) Inductive arguments are always valid; deductive arguments are based on
probability.
c) Inductive arguments rely on logic; deductive arguments rely on assumptions.
d) Deductive arguments guarantee conclusions; inductive arguments suggest
probable conclusions.
Answer: d) Deductive arguments guarantee conclusions; inductive arguments suggest
probable conclusions.
7) Which of the following is a valid deductive argument?
a) All birds can fly. Penguins are birds. Therefore, penguins can fly.
b) All mammals have lungs. Dolphins are mammals. Therefore, dolphins have lungs.
c) Some cats are black. Felix is a cat. Therefore, Felix is black.
d) Most students study hard. Sarah is a student. Therefore, Sarah studies hard.
Answer: b)All mammals have lungs. Dolphins are mammals. Therefore, dolphins have
lungs.
8) What is the term for a deductive argument with true premises and a true
conclusion?
a) Invalid
b) Probable
c) Sound
d) Plausible
Answer: c) Sound
9) Which of the following is a common form of inductive reasoning?
a) Mathematical proof
b) Categorical syllogism
c) Modus tollens
d) Statistical generalization
Answer: d) Statistical generalization
10) Which application of reasoning would most likely involve inductive arguments?
a) Solving a math problem
b) Drawing conclusions from formal definitions
c) Predicting the weather based on past data
d) Determining legal consequences based on laws
Answer: c) Predicting the weather based on past data
Short Answer Questions
1. Define a deductive argument and provide an example.
2. What are the key components of a deductive argument?

60
 3. Explain the difference between valid and invalid deductive arguments.
4. What are the characteristics of inductive arguments?
5. How can inductive reasoning be applied in everyday life?

Subjective Questions
1. Compare and contrast inductive and deductive arguments by giving examples of
each.
2. Explain the concept of validity and soundness in deductive arguments. Why are
they important?
3. Discuss the limitations of inductive reasoning in drawing conclusions.
4. Analyze a real-life scenario where both inductive and deductive reasoning are
used together.
5. How does understanding reasoning types improve decision-making in everyday
life?

61
Activities
1. Write ten arguments and classify each as inductive or deductive and explain the
reasoning.

2. Observe your surroundings and write down three examples of inductive reasoning
and three examples of deductive reasoning. Share your observations on the AIOU
LMS/MS Teams/LMS discussion forum.

3. Provide a weak inductive argument and suggest additional premises or evidence to

make it stronger.

4. Select a complex argument and break it into premises, reasoning, and conclusion,
mapping how each connects.

5. Describe a recent decision they made and analyze whether they used inductive or
deductive reasoning.

6. Select an advertisement with a claim and determine if the argument is deductive or

inductive and assess its validity or strength.

7. Describe a situation in your personal or professional life where inductive reasoning

helped you to make a decision.

8. Describe a situation where deductive reasoning was essential to solving a problem

or reaching a conclusion.

9. Find an example of a deductive argument in a news article, book, or other media.

Analyze its premises and conclusion for validity and soundness.

10. Find an example of an inductive argument in a scientific study, business report, or

other source. Evaluate its strength and the likelihood of the conclusion being true.

REFERENCES

Briggs, B.(2015). Using and understanding Mathematics: A quantitative reasoning

approach. USA: Pearson Education Inc.

Cottrell, S. (2015). Critical Thinking Skills. New York: Palgrave Macmillan Ltd.

Starkey, L. (2004). Critical thinking skills success in 20 minutes a day. New York:
Learning Express, LLC.

62
Unit–3

EVALUATING ARGUMENTS

Written by: Dr. Mubeshera Tufail

Reviewed by: Dr. Misbah Javed

63
 INTRODUCTION

This unit explains two types of arguments: inductive and deductive. Understanding
these arguments helps in thinking clearly and making better decisions. Inductive arguments
use evidence to make guesses, while deductive arguments provide clear reasoning to reach
definite conclusions.
This unit also focuses on fallacies, which are common mistakes in arguments. Learning
about fallacies is important for critical thinking. It helps us avoid being misled by false
arguments and improves how we evaluate information. Recognizing fallacies also makes
our communication stronger and more convincing.
By learning these skills, we can analyze arguments better and make informed decisions.
This connects with quantitative reasoning as it improves how we interpret data and draw
logical conclusions. Let us recall the definition of inductive and deductive arguments from
the previous unit.

 An inductive argument makes a case for a general conclusion from premises

that are more specific.
 A deductive argument makes a case for a specific conclusion from premises
that are more general.

Learning Outcomes
At the end of the unit, you should be able to:
1. Differentiate between inductive and deductive arguments.
2. Evaluate the strength of inductive arguments based on the evidence provided.
3. Assess the validity and soundness of deductive arguments.
4. Recognize various types of logical fallacies in arguments and understand why
they undermine the argument's validity.
5. Use the principles learned to critically analyze arguments in various contexts,
ensuring a more rigorous and logical approach to reasoning.
6. Construct the arguments in a clear, logical, and persuasive manner while
avoiding common logical fallacies.

64
3.1 Evaluating Deductive Argument
Assessing a deductive argument involves answering two crucial questions:
1. Does the conclusion necessarily follow from the premises?
This question pertains to the validity of the argument. A deductive
argument is considered valid if the conclusion logically follows from the
premises, without any logical errors. Validity is solely concerned with the
logical structure of the argument and does not involve personal judgment
or the truth of the premises or conclusion.
2. Are the premises true?
This question addresses the truthfulness of the premises. To be sure that
the conclusion of a deductive argument is true, it is necessary for all
premises to be true. If any premise is false, it undermines the reliability of
the argument.
If a deductive argument is both valid and has true premises, then it is sound.
Soundness represents the highest level of reliability for a deductive argument
because, theoretically, it guarantees the truth of the conclusion. However,
soundness may still involve some level of personal judgment, particularly if the
truth of the premises is debatable.
Argument 1 (Valid Deductive Argument)
Premise 1: All humans are mortal.
Premise 2: Person A is a human being.
Conclusion: Therefore, Socrates is mortal.

Argument 1 is valid because the conclusion logically follows from the premises.
If all humans are mortal (Premise 1) and person A is a human being (Premise 2),
then it logically follows that person A is mortal (Conclusion). The conclusion
necessarily follows from the premises, demonstrating the validity of the
argument.

Argument 2 (Truthfulness of the Premises)

Premise 1: All birds are mammals.
Premise 2: A sparrow is a bird.
Conclusion: Therefore, a dog is a mammal.

Argument 2 may appear valid because the conclusion follows logically from the
premises. However, it is not sound because Premise 1 ("All birds are mammals")
is false. In reality, birds are a distinct class of animals, separate from mammals.
Therefore, even though the argument may seem logically consistent, it leads to a
false conclusion due to the falsehood of one of the premises.

65
3.2 Evaluating Inductive Argument
Evaluating an inductive argument involves two factors: strength and truth.
Inductive arguments are evaluated based on their strength. A strong argument
makes the case for its conclusion seem quite convincing, even though it does not
prove the conclusion true. A weak argument is one in which the premises do not
seem to lend much support to the conclusion. Evaluating the strength of an
argument involves personal judgment.
The strength of an inductive argument does not necessarily correlate with the
truth of its conclusion. Argument 1, despite being strong, had a false conclusion.
The rockets launched into the space do not support this conclusion. Conversely, a
weak argument may have a true conclusion, but it remains weak due to insufficient
support. Argument 2 is a weak argument because the evidence provided in the
premises is not sufficient to conclude that it will definitely rain today. However,
the conclusion may be true if it rains later in the day.
Examples

Argument 1 (strong inductive argument with false conclusion)

Premise: Airplanes take off into the sky but eventually land back on the ground.
Premise: Kites flown into the air eventually descend back to the ground.
Premise: Helium balloons released into the air eventually fall back down to the
ground.
Premise: Paper airplanes thrown into the air eventually land on the ground.
Conclusion: Anything that ascends into the air will eventually descend back
down.

Argument 2 (weak inductive argument with true conclusion)

Premise: The weather forecast predicted rain for today.
Premise: Dark clouds have been looming overhead all morning.
Conclusion: It will rain today.

Key Points
 Evaluating an Inductive Argument: An inductive argument cannot prove
its conclusion true, so it can be evaluated only in terms of its strength. An
argument is strong if it makes a compelling case for its conclusion. It is weak
if its conclusion is not well supported by its premises.
 Evaluating a Deductive Argument: There are two criteria when evaluating
a deductive argument. The argument is valid if its conclusion follows
necessarily from its premises, regardless of the truth of the premises or
conclusions. The argument is sound if it is valid and its premises are all true.

66
 3.3 Logical Fallacies in Deductive Reasoning
An invalid argument is one that contains one or more errors, which can fall into the following
categories: factual error and logical fallacy.
(i) Factual Errors:
Premise Error: One or more premises are not true. If the premises are false, the
conclusion cannot be logically supported.
Conclusion Error: The conclusion does not logically follow from the premises, even if
the premises are true.
(ii) Logical Errors (Fallacies): A fallacy is an error in reasoning that renders an
argument logically invalid. Even if the premises are true, the fallacy prevents
the argument from being logically sound.

2.4.3 Major Types of Logical Fallacies in Deductive Arguments

There are four major logical fallacies in deductive arguments.
1. Slippery Slope
Slippery slope fallacy occurs when an argument suggests that a minor action
will lead to major and often ludicrous consequences. The conclusion takes
the premises to an extreme.
For example:
Premise 1: If we allow students to redo assignments, they'll expect to redo every
assignment.
Premise 2: If they redo every assignment, they'll never learn responsibility.
Conclusion: Therefore, we shouldn't allow students to redo assignments.

2. False Dilemma
The false dilemma fallacy, also known as the either-or fallacy, presents only two
options in its major premise when there are actually more possibilities.
Example:
Premise 1: You are either with us, or you are against us.
Premise 2: You are not with us.
Conclusion: Therefore, you are against us.
This ignores any middle ground or alternative options.

3. Circular Reasoning
Circular reasoning, or begging the question, occurs when the argument's
conclusion is essentially the same as one of its premises, often restated in
different terms.
Example:
Premise 1: The Bible is true because it is the word of God.
Premise 2: The Bible says God exists.

67
 Conclusion: Therefore, God exists.
Here, the argument assumes what it is trying to prove, leading to no new
information or proof.

4. Equivocation
Equivocation involves using a word with multiple meanings and switching between
those meanings within the argument, leading to a misleading or invalid
conclusion. Example:
Premise 1: A feather is light.
Premise 2: What is light cannot be dark.
Conclusion: Therefore, a feather cannot be dark.
In this case, "light" is used in two different senses—weight and brightness—which
makes the argument invalid.

Importance of Recognizing Fallacies

Recognizing these fallacies is crucial as they can make arguments seem
convincing despite their logical flaws. By identifying these errors, one can
avoid being misled or persuaded by faulty logic, ensuring a clearer and more
accurate understanding of the arguments presented.
Activity 1: Identifying Fallacies in Arguments
Identify logical fallacies in given arguments. Identify the fallacy in each
argument and explain why it is a fallacy.
i. Argument: "If we ban smoking in public places, next we'll be banning
cars because they pollute the air. Then we'll ban factories, and
eventually, everything will be banned." (Slippery Slope)
ii. Argument: "You're either a cat person or a dog person." (False
Dilemma)
iii. Argument: "The best way to prevent theft is to install security cameras
because security cameras are the best theft deterrent." (Circular
Reasoning)
iv. Argument: "The sign said 'Fine for Parking Here,' so since it was fine,
I parked there." (Equivocation)

Activity 2: Fallacy Detective

Find a newspaper or magazine article or watch the news on TV that contains
quotes from one or more politicians. Do any of them use logical fallacies in
their arguments? If so, which ones?
Activity 3: Creating Fallacious Arguments and Correcting it
Create arguments that contain logical fallacies (Slippery slope, False
Dilemma, Circular Reasoning and Equivocation), and then make it a valid
argument after corrections.
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2.5 Logical Fallacies in Inductive Arguments
Inductive reasoning involves making generalizations from specific instances. There
are some fallacies in inductive arguments, as given below:
2.5.1 Hasty Generalization: Making a broad conclusion based on a small sample
size. Example: "My friend got food poisoning from that restaurant, so all their
food must be unsafe."

A hasty generalization takes the following form:

1. A very small sample A is taken from population B.
2. Generalization C is made about population B based on sample A.

"I met three people from city A who were rude, so everyone from city A must
be rude."

Common Causes of Hasty Generalizations

Common causes of hasty generalizations are bias or prejudice, and
negligence or laziness. Bias or prejudice is drawing conclusions based on
personal biases or prejudices rather than objective evidence. For example, A
person concludes that all members of a certain ethnic group are lazy because
they had a negative encounter with one individual. Negligence or Laziness
means making broad generalizations due to a lack of effort in obtaining
sufficient evidence or considering alternative perspectives. For example,
assuming that all students from a particular school are academically gifted
based on the performance of a few exceptional students.
In order to avoid hasty generalizations, seeking better evidence and avoiding
unwarranted conclusions may be helpful. Seeking better evidence means
striving to gather more diverse and representative evidence to support your
conclusions instead of relying on a small or biased sample. For example,
conducting a comprehensive survey or study to obtain a larger and more varied
sample size. Avoiding unwarranted conclusions means refraining from making
sweeping generalizations when the available evidence is insufficient or biased.
For example, recognizing the limitations of personal experiences and biases
and seeking out additional data or perspectives before drawing conclusions.
2.5.2 Chicken and Egg Fallacy (Causal Oversimplification): It involves
incorrectly assuming a causal relationship between two events without
sufficient evidence to establish which one causes the other. For example,
"Poverty causes crime because poor neighborhoods have higher crime rates.
However, it is also possible that high crime rates contribute to poverty."

Chicken and egg is a fallacy that has the following general form:
69
 1. A and B regularly occur together.
2. Therefore, A is the cause of B.

The chicken and egg fallacy occurs when a causal relationship is assumed
between two events solely based on their correlation, without considering other
potential explanations or gathering adequate evidence to support the claim. For
example, high ice cream sales (A) are correlated with an increase in drowning
incidents (B) during the summer months. It is a fallacious conclusion to assume
that high ice cream sales cause an increase in drowning incidents because they
occur together. However, the increase in both ice cream sales and drowning
incidents during the summer months may be attributed to a common factor,
such as hot weather, which leads people to consume more ice cream and also
increases the likelihood of swimming and water-related accidents.
In order to mitigate the Chicken and Egg Fallacy, considering alternative
explanations may be helpful such as exploring other factors or variables that
could explain the observed correlation between A and B. Another strategy is
to seek additional evidence i.e., gather more data or conduct further research
to determine if there is a causal relationship between A and B or if they are
merely correlated by coincidence. Another suggestion is to refrain from
jumping to conclusions about causation based solely on correlation without
sufficient evidence.

2.5.3 Post Hoc, Ergo Propter Hoc (After This, Therefore Because of This): It
involves assuming that because one event precedes another, it must have
caused the second event. For example, "I wore my lucky socks before every
basketball game, and my team won. Therefore, my lucky socks must have
caused the victories."

2.5.4 Composition Fallacy: It involves assuming that what is true for the parts must
also be true for the whole. For example, "Each player on the team is an
excellent athlete, so the team as a whole must be unbeatable." However, team
dynamics and strategy also play a significant role.
It assumes that:
1. Since all of the parts of the machine (A) are lightweight (B),
2. Therefore, the machine as a whole (C) is lightweight (B).

“Because each player is excellent, the entire team must also be excellent.” It
ignores other factors such as teamwork, strategy, or coaching.
“The human body is made up of atoms, which are invisible. Therefore, the
human body is invisible.”
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 This fallacy arises when the qualities of the parts are incorrectly assumed
to apply to the entire whole without sufficient justification. It overlooks the
possibility of other factors or properties influencing the overall nature of the
whole entity. To determine whether an inference from the properties of the
parts to the properties of the whole is fallacious, it is crucial to assess the
justification for the inference.
In order to mitigate the composition fallacy, consider additional factors i.e.,
evaluate other factors or attributes that may influence the overall nature or
characteristics of the whole entity. Another tip is to assess the justification for
inference i.e., determine whether there is sufficient justification for inferring
that the qualities of the parts necessarily apply to the whole. Avoiding
overgeneralization may be helpful i.e., refrain from making broad
generalizations about the entire entity based solely on the qualities of its
individual components.

By engaging in the activities for inductive reasoning, you can practice recognizing
and forming inductive arguments, and understand the strengths and limitations of
this type of reasoning.
Activity 1: Observation and Generalization
Prepare a set of specific observations about a particular animal and draw a general
conclusion. For example,
"Every swan I've seen is white. What can you conclude about the color of swans?"

Activity 2: Comparative Arguments

Considering two scenarios or items, compare them and draw a conclusion. For
example, "This laptop has a high-performance processor and long battery life, similar
to the previous model. What can you infer about the new model's performance?"

Activity 3: Causal Arguments

Prepare a cause-and-effect scenario and determine the likely outcome. For example,
"Every time the temperature drops below freezing, the pond freezes over. It is
predicted to drop below freezing tonight. What is the likely outcome for the pond?"

Pay attention and evaluate your skills in inductive reasoning. Are you using common
sense and/or past experience for inductive arguments? Have you noticed a key
difference, or compared two similar events for it? Give an example of when and how
you use it.

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Summary
This unit explains two types of reasoning: inductive and deductive arguments. Inductive
arguments use evidence to make guesses, while deductive arguments give clear reasoning
to reach a definite answer. The unit also talks about common mistakes in reasoning, called
fallacies. These mistakes include making broad claims with little evidence, confusing cause
and effect, and thinking one event causes another just because it happened first. Other
errors include thinking what is true for parts is true for the whole, assuming one step leads
to extreme outcomes, and giving only two choices when there are more.
Some arguments go in circles or use tricky words to confuse people. Learning about
these fallacies helps in spotting mistakes, thinking clearly, and making better decisions in
daily life. This knowledge helps in understanding and building stronger arguments.
Multiple Choice Questions
1. Which of the following is an example of the slippery slope fallacy?
a. You are either with us, or you are against us.
b. The sun rises in the east because it has always risen in the east.
c. If we allow students to use calculators, they will forget how to do basic
math.
d. If it rains tomorrow, the ground will be wet.
Answer: c) If we allow students to use calculators, they will forget how to do
basic math.
2. A false dilemma is a fallacy that:
a) Assumes a cause-and-effect relationship without evidence.
b) Presents only two options when more exist.
c) Uses vague language to mislead.
d) Draw a conclusion based on insufficient data.
Answer: b) Presents only two options when more exist.
3. Which fallacy occurs when the conclusion is assumed in the premise?
a) Circular reasoning
b) Hasty generalization
c) Composition fallacy
d) Equivocation
Answer: a) Circular reasoning
4. What does the post hoc fallacy imply?
a) If two events occur together, one must cause the other.
b) A generalization is made based on insufficient evidence.
c) A conclusion is drawn by ignoring evidence.
d) One event caused another just because it occurred earlier.
Answer: d) One event caused another just because it occurred earlier.
5. Which is an example of hasty generalization?

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 a) The earth revolves around the sun because it always has.
b) My friend got sick after eating at that restaurant. All their food must be
bad.
c) A square has four sides; hence, every four-sided shape is a square.
d) If you do not study, you will fail every test in your life.
Answer: b) My friend got sick after eating at that restaurant. All their food must
be bad.
6. What type of fallacy uses ambiguous language to mislead?
a) Equivocation
b) Composition fallacy
c) Circular reasoning
d) Slippery slope
Answer: a) Equivocation
7. "If we allow one student to break the rules, soon no one will follow them." It is
an example of:
a) Chicken and egg fallacy
b) Slippery slope fallacy
c) Post hoc fallacy
d) Hasty generalization
Answer: b) Slippery slope fallacy
8. What is the primary problem with composition fallacy?
a) It assumes correlation implies causation.
b) It oversimplifies complex causal relationships.
c) It uses vague terminology to confuse.
d) It assumes what is true for the part is true for the whole.
Answer: d) It assumes what is true for the part is true for the whole.
9. Which fallacy suggests "after this, therefore because of this"?
a) Hasty generalization
b) Post hoc
c) False dilemma
d) Circular reasoning
Answer: b) Post hoc
10. The chicken and egg fallacy often involves:
a) Misunderstanding the sequence of cause and effect.
b) Making assumptions based on limited observations.
c) Presenting only two extreme choices.
d) Repeating the conclusion in the premise.
Answer: a) Misunderstanding the sequence of cause and effect

73
Short-Answer Questions
1. Define a logical fallacy and explain why it weakens an argument.
2. What is the difference between a hasty generalization and a composition
fallacy?
3. Provide an example of circular reasoning from daily life.
4. Why is it important to recognize logical fallacies in arguments?
5. Explain post hoc or ergo propter hoc fallacy. fallacy of post hoc, ergo propter
hoc.
Subjective Questions
1. Discuss the importance of identifying fallacies in both deductive and inductive
reasoning.
2. Write a detailed explanation of the slippery slope fallacy with two real-life
examples.
3. Compare and contrast the fallacies of false dilemma and equivocation. Provide
examples for each.
4. How can the chicken and egg fallacy affect decision-making processes in daily
life?
5. Analyze the implications of relying on hasty generalizations when forming
public policies.
Activities
1. Write an example of an inductive and a deductive argument.
2. Search online for ten short statements and identify whether each statement
contains a fallacy. If yes, they should name the fallacy and explain why.
3. Find examples of logical fallacies from news articles, advertisements, or social
media and describe the fallacy, explain its type, and discuss its potential impact.
4. Search online for two short argumentative paragraphs: one containing logical
fallacies and one without. Identify and critique the fallacies in the flawed
argument and explain why the other argument is logically valid.
5. Select a topic and prepare arguments for and against a proposition. It must be
ensured that your arguments avoid logical fallacies and critique opposing
arguments for any fallacies.
6. Create a list of fallacy names and a separate list of their examples or definitions,
then match the correct fallacy to its description or example (match the column
activity).
7. Search online for an argument with logical fallacies and rewrite the argument to
make it logically sound.

74
8. Identify a hasty generalization you have encountered recently. How could this
argument be strengthened with better evidence?
9. Describe a situation where you might encounter the chicken-and-egg fallacy.
How can you avoid making this error in your reasoning?
10. Evaluate the strength of the following inductive argument: "Every swan I have
seen is white, so all swans must be white".
11. Evaluate the validity and soundness of this argument: "All mammals have
lungs. A whale is a mammal. Therefore, a whale has lungs."
12. Reflect on a recent argument you encountered (in media, conversation, etc.).
Were there any fallacies present? If so, identify them and explain their impact
on the argument.
13. Describe a situation where you identified a fallacy in someone else's argument.
How did you address it, and what was the outcome?
14. Read the given scenarios carefully and complete the given task.
i. In a debate about climate change, one participant says, "If you care about
the environment, you must support our new policy. If you don't support
our policy, you clearly don't care about the environment." Identify the
fallacy in this argument and explain why it undermines the argument's
validity.
ii. A friend tells you, "Every time I wash my car, it rains the next day.
Therefore, washing my car causes it to rain." Evaluate this argument.
What type of reasoning is being used, and what fallacy, if any, can you
identify?
iii. During a team meeting, a colleague argues, "We should adopt the new
software because our competitor uses it, and they are very successful."
Analyze this argument. Identify any fallacies present and suggest a more
logical way to support the decision to adopt the new software.
iv. In a discussion about school uniforms, someone argues, "We should have
school uniforms because we've always had them, and changing them
would be too much trouble." Identify the fallacy in this argument. How
would you respond to promote a more productive discussion?
v. You read an article claiming, "Studies show that people who drink coffee
live longer. Therefore, drinking coffee will ensure you live a long life."
Evaluate the inductive argument presented in the article. Discuss any
potential fallacies and the overall strength of the argument.

75
 REFERENCES

Briggs, B.(2015). Using and understanding Mathematics: A quantitative reasoning

approach. USA: Pearson Education Inc.

Cottrell, S. (2015). Critical Thinking Skills. New York: Palgrave Macmillan Ltd.

Starkey, L. (2004). Critical thinking skills success in 20 minutes a day. New York:
Learning Express, LLC.

76
Unit–4

SETS AND VENN DIAGRAMS

Written by: Dr. Mubeshera Tufail

Reviewed by: Dr. Misbah Javed

77
 INTRODUCTION

This unit introduces the basic concepts of sets, propositions, and their relationships using
Venn diagrams. Sets are groups of distinct objects, and studying them helps explain many
mathematical ideas. Propositions are statements about sets and their relationships. Venn
diagrams, created by John Venn, are a simple way to visually show how sets overlap or
relate.
By using numbers in Venn diagrams, learners will see how to analyze and interpret data
more easily. Practical examples, like understanding course enrollments or employee skills,
show how Venn diagrams help organize and explain complex information. These topics
are closely linked to quantitative reasoning, as they help analyze data, solve problems, and
understand patterns in real-life situations. These topics are important for understanding
mathematics, logic, computer science, and data analysis.

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Learning Outcomes
After the successful completion of the unit, students would be able to::
1. Understand the set and its types.
2. Interpret Comprehend the nature of propositions, particularly categorical
propositions, and their role in asserting relationships between sets.
3. Use Venn diagrams to visually represent relationships between sets, including
disjoint sets, and subsets.
4. Analyze the relationships between sets using Venn diagrams, while understanding
the implications of overlapping and non-overlapping regions.
5. Integrate numerical data into Venn diagrams to provide a detailed analysis of how
elements are distributed among sets.
6. Apply the concepts of sets, propositions, and Venn diagrams to real-life scenarios
to organize and interpret complex information.
7. Draw logical conclusions from the visual and numerical data represented in Venn
diagrams, enhancing problem-solving and analytical skills.
8. Share clear findings from Venn diagram analysis, both verbally and in written
form.

4.1 Defining Proposition, Set and Venn Diagram

4.1.1 Propositions
A proposition is a declarative statement that is either true or false. In logic,
propositions are the basic units that can be combined using logical
connectives (such as AND, OR, NOT) to form more complex statements.
4.1.2 Sets
A set is a collection of objects; the individual objects are the members of
the set. We write sets by listing their members within a pair of braces {}. If
there are too many members to list, we use three dots (called ellipsis), “ . . .
,” to indicate a continuing pattern. Sets can be represented using curly
brackets, e.g., A={1,2,3} and B= {2,4,6,7,8,…}. A few more examples are
given below:

 Set of Knives: {Chef knife, Paring knife, Bread knife, Utility knife}

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  Set of Spoons: {Teaspoon, Tablespoon, Dessert spoon, Soup spoon}
 Set of Writing Instruments: {Pen, Pencil, Marker, Crayon}
 Set of Notebooks: {Math notebook, Science notebook, English
notebook, History notebook}
 Set of Cardio Exercises: {Running, Cycling, Swimming, Jumping
rope}
 Set of Strength Exercises: {Push-ups, Squats, Deadlifts, Bench
press}
 Set of Personal Gadgets: {Smartphone, Laptop, Tablet,
Smartwatch}
 Set of Home Appliances: {Refrigerator, Microwave, Washing
machine, Dishwasher}
 Set of Genres: {Fiction, Non-fiction, Mystery, Science Fiction}
(Please visit quantitative reasoning-I course about the detailed concept of
sets.)
4.1.3 Venn Diagrams
A Venn diagram is a visual representation of mathematical or logical
relationships between different sets. Each set is represented by a circle, and
the position and overlap of the circles indicate the relationships between the
sets. Venn diagrams are used to illustrate concepts such as unions,
intersections, and complements. John Venn (1834–1923), the English
logician, revolutionized the visualization of set relationships through his
invention of Venn diagrams. These diagrams, employing circles to
represent sets, offer an intuitive means of understanding complex
relationships.
In the context of sets like "whales" and "mammals," Venn diagrams
illustrate the notion of a subset: where every member of one set is also a
member of another. In this case, as all whales are mammals, the circle
representing whales is entirely within the circle representing mammals. The
diagram delineates three regions: the interior of the whale's circle
representing all whales, the area outside the whale's circle but inside the
mammal circle representing mammals excluding whales, and the space
outside the mammal circle representing non-mammals.
When considering sets like "dogs" and "cats," Venn diagrams depict
disjoint sets i.e., sets with no common elements. Dogs and cats, as separate
categories of pets, are depicted as circles that do not intersect. This visual
method aids in grasping the relationships between sets and their elements,

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facilitating logical reasoning and problem-solving across various
disciplines.

Figure 4.2 Venn diagram for whales and mammals

Figure 4.3 Venn diagram for dogs and cats

Let us see another example of two sets: women and nurses. Because of the
overlapping region, this diagram has four distinct areas:
 The overlapping region: This area represents individuals who are
both women and nurses—female nurses.
 The non-overlapping region of the nurses' circle: This area
signifies nurses who are not women—male nurses.
 The non-overlapping region of the women's circle: This area
denotes women who are not nurses.
 The region outside both circles: From the context, this area is
interpreted to represent individuals who are neither nurses nor
women—men who are not nurses.

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 This breakdown helps in understanding how Venn diagrams can visually
separate and categorize different subsets and intersections within a given
universal set.

Figure 4.4 Venn diagram for women and nurses

Note that the sizes of the regions in a Venn diagram are not important.
For instance, a small overlapping region does not imply that female nurses
are less common than male nurses. In fact, some regions might even have
no members. More broadly, overlapping circles are used in Venn diagrams
to show that two sets might have members in common, regardless of the
actual number of members in each subset.

4.1.4 Relationship Among Proposition, Set and Venn Diagram

(i) Propositions and Sets: Propositions can describe conditions that
define a set. For example, we have two propositions P(x): “x is an
even number” and Q(x): “x is a number greater than 5”. Two sets
based on these propositions can be defined as given below:
Set A defined by the proposition P(x): A= {2, 4, 6, 8, 10…..}
Set B defined by the proposition Q(x): B= {6, 7, 8, 9, 10, 11,…}

Categorical propositions assert a particular relationship between

two categories or sets. For example, the proposition "all whales are
mammals" makes a specific claim that the set of whales is a subset
of the set of mammals. Like all propositions, categorical
propositions must be structured as complete sentences. They also
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 possess another important general feature: one set appears in the
subject of the sentence, and the other appears in the predicate. For
example, in the proposition "all whales are mammals," the set of
whales is the subject (S) set, and the set of mammals is the predicate
(P) set. We can write it as given below:

All S are P, where S = whales and P = mammals.

Categorical propositions come in the following four standard forms

(figure 4.5).

Figure 4.5 Four Forms of Categorical Propositions

(ii) Sets and Venn Diagrams: Venn diagrams are tools to visualize
relationships between sets. For instance, if A and B are set, their
union A∪B, intersection A∩B, and difference A−B can be visually
represented using overlapping circles. Let us see it for the above-
mentioned sets A and B.

Intersection A∩B (includes numbers that are even and greater than
5)
A∩B = {6, 8, 10, 12,…}

Union A∪B (elements that are either even or greater than 5)

A∪B = {2, 4, 6, 7, 8, 9, 10, 11,…}

Difference A-B (elements that are even but not greater than 5)
A-B= {2, 4}

Propositions and Venn Diagrams: Propositions can be represented

using Venn diagrams when they describe conditions that involve
sets. For example, if the proposition "An element is in A or B"
corresponds to the union A∪B, which can be depicted in a Venn

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diagram showing the areas covered by both sets. The difference A-
B, explained in the previous point [A-B= {2, 4}], can be represented
by the following diagram.

S
e 2 S
t 4 e
A t
B

The four categorical propositions (figure 4.5) can be depicted

through Venn diagrams as given below.

All S are P.

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No S are P.

Some S are P.

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 Some S are not P.

Let us see Venn diagram for two propositions.

a. Some birds can fly.
b. Elephants never forget

Logical operations on propositions can be represented similarly.

The logical AND (conjunction) corresponds to the intersection of
sets, OR (disjunction) corresponds to the union, and NOT (negation)
corresponds to the complement of a set.

Example:

Propositions
Consider the following propositions related to items you might want to
buy:
P(x): "x is a fruit."
Q(x): "x is on sale."

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 Sets Defined by Propositions
Using these propositions, we can define two sets:
Set 𝐴 defined by the proposition P(x): A={apple, banana, orange, grape,
peach}
Set 𝐵 defined by the proposition Q(x): B={banana, orange, bread, milk,
peach}

Now, we can consider various set operations based on these propositions:

Intersection: A∩B (items that are both fruits and on sale)
A∩B={banana, orange, peach}
Union: A∪B (items that are either fruits or on sale)
A∪B={apple, banana, orange, grape, peach, bread, milk}
Difference: A−B (items that are fruits but not on sale)
A−B={apple, grape}
Visualization with a Venn Diagram
Here’s how you can visualize these sets with a Venn diagram:

4.2 Venn diagrams with Three sets

Venn diagrams are particularly useful for dealing with three sets that may overlap
one another. For example, suppose you are conducting a study to learn how teenage
employment rates differ between boys and girls and between honor students and
others. For each teenager in your study, you need to record the answers to these three
questions:
A. Is the teenager a boy or a girl?

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B. Is the teenager an honor student or not?
C. Is the teenager employed or not?

Figure 4.6 shows a Venn diagram that can help you organize this information.
Notice that it has three circles, representing the sets boys, honor students, and
employed. Because the circles overlap one another, they form a total of eight regions
(including the region outside all three circles).
1. Region inside all three circles: Represents boys who are honor students and
are employed.
2. Region inside the circles for boys and honor students, but outside the
employed circle: Represents boys who are honor students but not employed.
3. Region inside the circles for boys and employed, but outside the honor
students circle: Represents boys who are employed but not honor students.
4. Region inside the circles for boys only: Represents boys who are neither
honor students nor employed.
5. Region inside the circles for honor students and employed, but outside
the boys circle: Represents honor students who are employed but not boys
(i.e., girls who are honor students and employed).
6. Region inside the honor students circle only: Represents honor students
who are not boys and not employed (i.e., girls who are honor students but
not employed).
7. Region inside the employed circle only: Represents those who are
employed but neither boys nor honor students (i.e., girls who are employed
but not honor students).
8. Region outside all three circles: Represents those who are neither boys,
honor students, nor employed (i.e., girls who are not honor students and not
employed).

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 Figure 4.6 Venn diagram with three sets

4.3 Venn diagrams with Numbers

Venn diagrams are primarily used to describe relationships between sets, such
as whether two sets overlap or whether overlapping sets share common members.
However, Venn diagrams can be even more useful when we add specific
information, such as the number of members in each set or overlapping region. The
following Figure 4.7 illustrates some of the ways in which Venn diagrams can be
used with numbers. It showed the number of classes with dogs and cats as their pets.
There were 10 students with cats, 7 students with dogs, 5 students with both dogs
and cats, and 8 students without any dogs and cats as pets.

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 Figure 4.7 Students with pets "cats" and "dogs"

Figure 4.8 added another set of “birds” as pets besides cats and dogs for a class
of 40 students. Try understanding its interpretation keeping in view the explanation
for figure 4.6.

Figure 4.8 Students with pets "cats", "birds" and "dogs"

Summary
In this unit, we explored the fundamental concepts of sets, propositions, and
their relationships using Venn diagrams. Venn diagrams are a powerful visual tool
that helps in understanding the relationships between different sets. These diagrams
use overlapping circles to depict how sets interact, whether they share common
elements, or if one set is a subset of another. We learned about sets, disjoint sets,

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 and categorical propositions (propositions that make specific claims about
relationships between sets using complete sentences, where one set is the subject
and the other is the predicate). Venn diagrams illustrate these relationships, focusing
on logical connections. Venn diagrams become more informative when specific
numerical data is added, allowing for a detailed analysis of how elements are
distributed among the sets. In summary, Venn diagrams are a versatile and intuitive
method for visualizing and analyzing the relationships between sets, enhancing our
ability to make sense of how different categories interact and helping us draw
meaningful conclusions from the data.

Multiple Choice Questions (MCQs)

1. What is a proposition in the context of sets and logic?
a) A logical statement that is either true or false.
b) A diagram representing elements.
c) A collection of unordered data.
d) A relationship between two variables.
Answer: a) A logical statement that is either true or false
2. Which of the following is NOT a component of a Venn diagram?
a) Circles
b) Overlapping regions
c) Labels for sets
d) Axes like in a graph
Answer: d) Axes like in a graph
3. What is the main purpose of using a Venn diagram?
a) To organize and visualize relationships between sets.
b) To perform calculations on numbers.
c) To list items in chronological order.
d) To create random groupings.
Answer: a) To organize and visualize relationships between sets.
4. A Venn diagram with three sets is typically represented using:
a) Two overlapping circles.
b) Three overlapping circles.
c) A single rectangle.
d) Four overlapping circles.
Answer: b) Three overlapping circles.
5. In a Venn diagram, shaded areas usually represent:
a) The universal set.
b) Specific relationships or subsets.
c) The complement of all sets.

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 d) Random elements of a set.
Answer: b) Specific relationships or subsets
6. Which term refers to the entire collection of elements under consideration?
a) Subset
b) Universal set
c) Complement
d) Null set
Answer: b) Universal set
7. What does the complement of a set (𝐴′) represent?
a) All elements in 𝐴
b) All elements not in 𝐴
c) All subsets of 𝐴
d) The total number of elements in A.
Answer: b) All elements not in 𝐴
8. Which of these is an example of a proposition?
a) x+y
b) "The sky is blue."
c) 2+2=4
d) Both b and c
Answer: d) Both b and c
9. What is the primary feature of a Venn diagram with numbers?
a) It uses numerical data to represent sets.
b) It avoids overlapping regions.
c) It excludes the universal set.
d) It only works for two sets.
Answer: a) It uses numerical data to represent sets
10. If a set has no elements, it is called:
a) A universal set
b) A null set
c) A complement set
d) An infinite set
e) Answer: b) A null set
Short-Answer Questions
1. Define a proposition and provide an example.
2. What is a universal set, and how is it represented?
3. Explain the difference between a set and a subset.
4. What is the complement of a set? Give an example.
5. What is a categorical proposition, and how does it relate to sets?

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 6. Give an example of a categorical proposition and represent it using set notation.
7. Describe the subject and predicate sets in the proposition: "All dogs are
mammals."
8. Explain the purpose of using Venn diagrams for sets.
9. Draw a Venn diagram representing the relationship between the sets A = {red,
blue} and B = {blue, green}.
Subjective Questions
1. Describe the regions in a Venn diagram and their significance in understanding
set relationships.
2. Discuss the importance of using Venn diagrams in organizing and representing
data.
3. Explain how propositions are used in logic and reasoning, with examples.
4. Describe the significance of the complement of a set in logical reasoning and set
theory.
5. Write a short essay on how set theory and Venn diagrams can help in solving real-
life problems, such as organizing groups or categories.
6. Illustrate with examples the difference between a null set and a universal set.

Activities
1) Write three propositions related to their daily lives and identify whether each
proposition is true or false and explain why.
2) Consider two categories (e.g., favorite hobbies and favorite foods) and draw a
Venn diagram to visually represent the relationship between these two categories.
3) Take a list of random items (e.g., animals, vehicles, fruits) and group them into
sets based on criteria they define (e.g., "Can Fly," "Lives in Water").
4) Enlist the items you see in a room and categorize them into "Electronic Devices"
and "Non-Electronic Devices", and then identify the complement of each set.
5) Create a short survey (e.g., "Do you prefer tea or coffee?"), collect responses from
family or friends and represent the data using a Venn diagram with numbers.
6) For a hypothetical scenario "Students who like sports vs. students who like
painting", create Venn diagrams to show possible relationships and analyze
overlaps.
7) Enlist various items in your surroundings and describe the characteristics of
possible sets they can form. For example, "All red objects" or "Items that weigh
less than 1 kg."
8) Think about examples where a null set might occur in their lives (e.g., "Students
in my class who can fly"), and explain why such sets exist and their significance.
9) Read the following scenarios and try answering the questions.

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 a. You are studying the dietary habits of individuals in a community. You
categorize individuals based on their dietary preferences such as vegetarian,
vegan, or omnivore, and whether they consume dairy products. Construct a Venn
diagram to represent the dietary preferences of individuals in the community,
considering their preferences for vegetarian, vegan, and omnivore diets. How
many individuals consume dairy products, and how many follow a vegetarian
diet?
b. You are analyzing the attendance records of students in a school. You categorize
students based on their participation in extracurricular activities, such as sports,
arts, and clubs. Using a Venn diagram, visualize the relationships between
students involved in sports, arts, and clubs. How many students are involved in
multiple activities, and how many are not involved in any extracurricular
activities?
c. You are conducting a survey to analyze the preferences of customers in a retail
store. You categorize customers based on their shopping habits, including whether
they prefer online shopping, in-store shopping, or both. Using Venn diagrams,
illustrate the different categories of customers and their preferences for online, in-
store, or both types of shopping.

REFERENCES

Briggs, B.(2015). Using and understanding Mathematics: A quantitative reasoning

approach. USA: Pearson Education Inc.

Cottrell, S. (2015). Critical Thinking Skills. New York: Palgrave Macmillan Ltd.

Starkey, L. (2004). Critical thinking skills success in 20 minutes a day. New York:
Learning Express, LLC.

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Unit–5

EXPONENTIAL GROWTH

Written by: Dr. Mubeshera Tufail

Reviewed by: Dr. Muhammad Tanveer Afzal

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 INTRODUCTION

In this unit, we will explore key concepts such as functions, independent and dependent
variables, and linear and exponential models. These ideas help us understand how different
factors or quantities are connected and how they change over time. Functions describe how
one variable influences another, while linear models focus on steady changes and
exponential models explain how things grow or shrink at a constant rate. By learning about
these models, students develop the skills to analyze data and recognize patterns in real-
world situations. These concepts are essential for quantitative reasoning, enabling learners
to make informed decisions, predict trends, and solve problems in various fields. This
knowledge equips them with the tools to deal with complex challenges and apply
mathematical thinking to everyday life.

Learning Outcomes
At the successful completion of the unit, the students would be able to:
1. Interpret the characteristics of linear and exponential growth.
2. Calculate doubling time and half-life.
3. Solve problems involving the growth and decay of populations and substances.
4. Explain the factors that influence population growth.
5. Explain the concept of carrying capacity and its determining factors.
6. Differentiate between exponential and logistic growth models.
7. Explain the purpose and application of logarithmic scales in measuring
phenomena (earthquake, sound and pH).
8. Calculate changes in sound intensity and understand their implications for
perceived loudness.
9. Use the pH scale to classify substances as acidic, neutral, or basic.
10. Describe the inverse square law and its relevance to phenomena such as sound
intensity.

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5.1 Introduction to Exponential Growth
Linear growth occurs when a quantity grows by the same absolute amount in
each unit of time. Exponential growth occurs when a quantity grows by the same
relative amount i.e., by the same percentage, in each unit of time. Linear growth is
growth that happens at the same rate of change. Every increase in x would bring
about the same increase in y. It is constant. With exponential growth, there is a
constant multiplier, so the growth rate is changing. In Figure 5.9, the blue straight
line represents linear growth and the red curved line represents exponential growth.

Figure 5.9 Comparison of Linear and Exponential Growth

Recall the concept of exponent from Quantitative Reasoning-I course (course

code: 8277). An exponent is a small number written above and to the right of a base
number, indicating how many times the base is multiplied by itself (e.g.,
23=2×2×2=8). Exponential growth builds on this concept, describing a situation
where a quantity increases rapidly by multiplying by a consistent factor over time.
For example, in exponential growth, if a population doubles every year starting at
100, after 1 year it is 100⋅21 =200, and after 2 years it is 100⋅22=400. Exponents
provide the foundation for understanding this rapid increase.

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Example of Exponential Growth
A city is growing at a rate of 1.6% per year. The initial population in 2010 is
P0=125,000. Calculate the city’s population over the next few years.
Solution
The relative growth rate is 1.6%. This means an additional 1.6% is added to 100%
of the population that already exists each year. This is a factor of 101.6%.

Population in 2010 = 125,000

Population in 2011 = 125,000(1.016)1=127,000

Population in 2012 = 127,000(1.016)=125,000(1.016)2=129,032

Population in 2013 = 129,032(1.016)=125,000(1.016)3=131,097

We can create an equation for the city’s growth. Each year the population is
101.6% more than the previous year.

P(t)=125,000(1+0.016)t

The city population shows an exponential growth trend.

Example of Exponential Growth

A city has declined in population at a rate of 1.6 % per year over the last 60 years.
The population in 1950 was 857,000. Find the population in 2014. (Wikipedia,
n.d.)

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Solution

P0=857,000

The relative growth rate is 1.6%. This means 1.6% of the population is subtracted
from 100% of the population that already exists each year. This is a factor of
98.4%.

Population in 1951 857,000(0.984)1=843,288

Population in 1952 843,228(0.984)=857,000(0.984)2=829,795

Population in 1953 828,795(0.984)=857,000(0.984)3=816,519

We can create an equation for the city’s growth. Each year the population is 1.6%
less than the previous year.

P(t)=857,000(1−0.016)t

So the population of the city in 2014, when t=64, is:

P(64)=857,000(1−0.016)64=857,000(0.984)64=305,258

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 Figure 5.10 Graph of city's population decline (declining exponential growth
model)

Example of Linear Growth

Dora has inherited a collection of 30 antique frogs. Each year she vows to buy
two frogs a month to grow the collection. This is an additional 24 frogs per year.
How many frogs will she have in six years? How long will it take her to reach 510
frogs?

The initial population is 0 P = 30 and the common difference is d = 24. The linear
growth model for this problem is:
P(t) = 30 + 24t

The first question asks how many frogs Dora will have in six years so, t = 6.

P(6) = 30 + 24(6) = 30 + 144 = 174 frogs

The second question asks for the time it will take for Dora to collect 510 frogs.
So, P(t) = 510 and we will solve for t.

510 = 30 + 24t

480 = 24t
100
 20 = t

It will take 20 years to collect 510 antique frogs.

Figure 5.11 Number of antique frogs accumulated over time (linear growth)

You can check more example here

https://math.libretexts.org/Bookshelves/Applied_Mathematics/Book%3A_College
_Mathematics_for_Everyday_Life_(Inigo_et_al)/04%3A_Growth/4.02%3A_Expo
nential_Growth, https://www.coconino.edu/resources/files/pdfs/academics/arts-
and-sciences/MAT142/Chapter_4_Growth.pdf and
https://math.libretexts.org/Bookshelves/Applied_Mathematics/Book%3A_College
_Mathematics_for_Everyday_Life_(Inigo_et_al)/04%3A_Growth/4.01%3A_Linea
r_Growth

The formula for Exponential Growth Model

P(t)=P0(1+r)t

P0 is the initial population.

r is the relative growth rate.
t is the time unit.
r is the positive if the population is increasing and negative if the population is
decreasing.

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 Formula for Linear Growth

Linear Growth Model: Linear growth begins with an initial population called P0.
In each time period or generation t, the population changes by a constant amount
called the common difference d. The basic model is:

P(t)= P0 + td

Key Facts about Exponential Growth

 Exponential growth leads to repeated doublings. With each doubling, the
amount of increase is approximately equal to the sum of all preceding
doublings.
 Exponential growth cannot continue indefinitely. After only a relatively
small number of doublings, exponentially growing quantities reach
impossible proportion.

5.2 Doubling Time and Half-Life

Understanding the concept of doubling time is crucial for comprehending
exponential growth. In essence, the doubling time represents the period it takes for
a quantity to double in size. This concept is applicable across various domains, from
finance to biology. Let us consider an example:
Initial population: 10,000
Doubling time: 10 years

Given that the population doubles every 10 years, we can calculate the population
at any given time using the formula:

Population = Initial Population × 2Time/Doubling Time

Using this formula:
 In 10 years (1 doubling time), the population will be 10,000 × 210/10 =
10,000×21 = 10,000×2= 20,000
 In 20 years (2 doubling times), the population will 10,000×220/10 = 10,000 ×
22 = 10,000 × 4 = 40,000
 In 30 years (3 doubling times), the population will be 10,000× 230/10 =
10,000 × 23= 10,000 × 8 = 80,000

It illustrates how the population grows exponentially over time with each
doubling period. The concept of doubling time provides a simple yet powerful tool
for understanding and predicting growth patterns in various contexts.

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5.3 Factors Affecting Population Growth

The world population growth rate is the difference between the birth rate and the
death rate:
Growth rate = birth rate – death rate

For example, for 2012 the global averages were 19 births per 1000 people and 8
deaths per 1000 people per year. Therefore, the population growth rate in 2012 was:

19 - 8 = 11 = 0.011 = 1.1%
1000 1000 1000

It is interesting to note that birth rates have significantly declined globally over
the past 60 years, which coincides with the period of the most substantial population
growth in history. Currently, birth rates around the world are at an all-time low.
However, the ongoing rapid population growth is primarily due to a more
pronounced decline in death rates.
Example:
In 1950, the world birth rate was 37 births per 1000 people and the world death
rate was 19 deaths per 1000 people. By 1975, the birth rate had fallen to 28 births
per 1000 people and the death rate to 11 deaths per 1000 people. Contrast the
overall growth rates in 1950 and 1975.

Solution
In 1950, the overall growth rate was

37 – 19 = 18 = 0.018 =
1.8%
1000 1000 1000
In 1975, the overall growth rate was:

28 – 11 = 17 = 0.017 =
1.7%
1000 1000 1000
Despite a significant fall in birth rates during the 25-year period, the growth rate
barely changed because death rates fell almost as much.

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5.3.1 Theoretical Models of Population Growth
Theoretical models of population growth therefore assume that the human population is
ultimately limited by the carrying capacity of Earth i.e., the number of people that Earth
can support. For any particular species in a given environment, the carrying capacity is the
maximum sustainable population. That is, it is the largest population the environment can
support for extended periods of time. Two important models for populations approaching
the carrying capacity are (1) a gradual leveling off, known as logistic growth, and (2) a
rapid increase followed by a rapid decrease, known as overshoot and collapse. Let us
investigate each model.

When the population is small relative to the carrying capacity, logistic growth is
exponential with a fractional growth rate close to the base growth rate r. As the population
approaches the carrying capacity, the logistic growth rate approaches zero. For example, if
the carrying capacity is 12 billion people, a logistic model assumes that the population
growth rate decreases as this number is approached. The fractional logistic growth rate at
any particular time depends on the population at that time, the carrying capacity, and the
base growth rate r.

Figure 5.12 contrasts logistic and exponential growth for the same base growth rate r.
In the exponential case, the growth rate stays equal to r at all times. In the logistic case, the
growth rate starts out equal to r, so the logistic curve and the exponential curve look the
same at early times. As time progresses, the logistic growth rate becomes ever smaller than
r, and it finally reaches zero as the population levels out at the carrying capacity.

Figure 5.12 Graph contrasts exponential growth with logistic

growth for the same base growth rate r

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 A logistic model assumes that the growth rate automatically adjusts as the
population approaches the carrying capacity. However, because of the astonishing
rate of exponential growth, real populations often increase beyond the carrying
capacity in a relatively short period of time. This phenomenon is called overshoot.
When a population overshoots the carrying capacity of its environment, a decrease
in the population is inevitable. If the overshoot is substantial, the decrease can be
rapid and severe—a phenomenon known as collapse. Figure 5.13 contrasts a logistic
growth model with overshoot and collapse. Overshoot and collapse characterize
many predator–prey populations. The population of a predator increases rapidly,
causing the prey population to collapse. Once the prey population collapses, the
predator population must also collapse because of a lack of food. Once the predator
population collapses, the prey population can begin to recover as long as it has not
gone extinct.

Figure 5.13 Contrast of logistic growth with overshoot and collapse

Given that the human population cannot grow exponentially forever, logistic
growth is clearly preferable to any kind of overshoot and collapse. Logistic growth
means a sustainable future population, while overshoot and collapse might mean the
end of our civilization.

5.3.2 Carrying Capacity

Understanding the concept of carrying capacity is crucial for assessing the
sustainability of population growth. Carrying capacity is the maximum population
size that an environment can sustain without detrimental effects on the environment
and its inhabitants. The key issue is whether the global population will stabilize or
exceed the carrying capacity, potentially leading to a severe population decline.

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 Key Factors Affecting Carrying Capacity:
i. Resource Usage: The rate of resource consumption significantly impacts
carrying capacity. Consumption rates vary greatly between countries. For
example, if the global population consumed resources at the same rate as the
average American, the carrying capacity would be lower compared to the rate of
resource consumption in Japan, which is about half that of the U.S.
ii. Environmental Impact Per Person: The environmental impact of each
individual influences carrying capacity. A higher average environmental impact
means a lower carrying capacity, as the environment's ability to support life is
diminished more quickly.
iii. Technological and Environmental Shifts: Advances in technology and
changes in the environment can alter the carrying capacity. Technological
innovations, such as fusion energy, could increase the carrying capacity by
providing abundant freshwater through desalination. Conversely, environmental
changes like global warming could decrease the Earth's ability to sustain human
life by affecting food production and other essential systems.
iv. System Complexity and Predictability: The Earth's complex system makes it
challenging to accurately predict carrying capacity. The numerous
interconnected factors and their potential changes add a level of unpredictability
to these estimates.

5.4 Logarithmic Scales: Earthquakes, Sounds and Acids

You might have heard earthquake strength described in magnitudes, sound
loudness in decibels, or the acidity of household cleaners in terms of pH. Each of
these measurement scales involves exponential growth, where each successive
number on the scale represents the same relative increase. For instance, a liquid with
a pH of 5 is ten times more acidic than one with a pH of 6. In this section, we will
delve into these three important scales. These are known as logarithmic scales,
which makes sense considering that logarithms are powers (see Figure 5.14).

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 Figure 5.14. Basic concept of the logarithm

5.4.1 The Magnitude Scale for Earthquakes

Earthquakes are a common occurrence for many people around the world (Figure
5.15 shows some great earthquakes around the world since 1960). In Pakistan,
northern and western parts of the country are the most earthquake-prone areas,
though earthquakes can happen almost anywhere. While most earthquakes are minor
and barely felt, severe earthquakes can cause the deaths of tens of thousands of
people. Table 5.2 shows the frequencies of earthquakes of varying strengths
according to standard categories defined by geologists.
Scientists measure the strength of earthquakes using the earthquake magnitude
scale, which is related to the energy released by an earthquake. Each step on this
scale represents approximately 32 times more energy than the previous step. For
example, a magnitude 8 earthquake releases 32 times more energy than a magnitude
7 earthquake.
Table 5.2 Earthquake categories and their frequency
Category Magnitude Approximate Number per year
(Worldwide Average Since 1900)
Great 8 and up 1
Major 7-8 18
Strong 6-7 120
Moderate 5-6 800
Light 4-5 6000
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 Minor 3-4 50,000
Very minor Less than 3 Magnitude 2-3: 1000 per day
Magnitude 1-2: 8000 per day

Figure 5.15 Great Earthquakes since 1960 around the world

(https://earthquake.usgs.gov/earthquakes/map)
Definition of Earthquake Magnitude Scale
The magnitude scale for earthquakes is defined so that each magnitude represents
about 32 times as much energy as the prior magnitude. More technically, the
magnitude, M, is related to the released energy, E, by the following equivalent
formulas:

The energy is measured in joules (joule is a unit to measure energy); magnitudes

have no units.

Earthquakes of the same magnitude can cause vastly different amounts of

damage depending on how their energy is released. Every earthquake releases some
energy into the Earth's interior, where it is relatively harmless, and some along the
Earth's surface, where it causes the ground to shake. A moderate earthquake that
releases most of its energy along the surface can cause more damage than a stronger
earthquake that releases most of its energy into the interior.

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 Deaths from earthquakes generally occur indirectly. Ground shaking can lead
to the collapse of buildings, making the worst earthquake disasters more likely in
regions where people cannot afford the high cost of earthquake-resistant
construction. Additionally, earthquakes can trigger other disasters, such as
landslides or tsunamis, which can cause further destruction and loss of life.
Example 1:
Using the formula for earthquake magnitudes, calculate precisely how much more
energy is released for each 1 magnitude on the earthquake scale. Also, find the
energy change for a 0.5 change in magnitude.

Solution
Let us look at the formula that gives the energy:

The first term, 2.5×104, is a constant number that is the same no matter what value
we use for M. The magnitude appears only in the second term, 101.5 M. Each time
we raise the magnitude by 1, such as from 5 to 6 or from 7 to 8, the total energy
E increases by a factor of 101.5. Therefore, each successive magnitude represents
101.5 ≈ 31.623 times as much energy as the prior magnitude. That is, each change
of 1 magnitude corresponds to approximately 32 times as much energy. Similarly,
a change of 0.5 magnitude corresponds to a factor of 101.5 × 0.5 = 100.75 ≈ 5.6 in
energy.

Example 2:
The 1989 San Francisco earthquake, in which 90 people were killed, had a
magnitude
7.1. Calculate the energy released, in joules. Compare the energy of this
earthquake
to that of the 2003 earthquake that destroyed the ancient city of Bam, Iran, which
had a magnitude of 6.3 and killed an estimated 50,000 people.

Solution
The energy released by the San Francisco earthquake was:
E= (2.5×104) ×
101.5M
= (2.5×104) ×
101.5×7.1
= 1.1×1015 joules

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 The San Francisco earthquake was 7.1 - 6.3 = 0.8 magnitude greater than the
Iran earthquake. It therefore released 101.5 × 0.8 = 101.2 ≈ 16 times as much energy.
Nevertheless, the Iran earthquake killed many more people because more
buildings
collapsed.

5.4.2 Measurement of Sounds in Decibels

The decibel scale is used to measure and compare the loudness of sounds. It is
defined such that a sound of 0 decibels (abbreviated 0 dB) represents the softest
sound audible to the human ear. Table 5.3 provides the approximate loudness of
various common sounds.
Table 5.3 Typical Sounds and Loudness in Decibels
Decibels Times as Loud as Example
Softest Audible Sound
140 1014 Jet at 30 meters
12
120 10 Strong risk of damage to the
human ear
100 1010 Siren at 30 meters
9
90 10 Threshold of pain for the human
ear
80 108 Busy street traffic
6
60 10 Ordinary conversation
4
40 10 Background noise in the average
home
20 102 Whisper
1
10 10 Rustle of leaves
0 1 Softest audible sound
-10 0.1 Inaudible sound

Decibel Scale for Sound

The loudness of a sound in decibels is defined by the following equivalent
formulas:

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Or

Example 1
Suppose a sound is 100 times as intense as the softest audible sound. What is its
loudness in decibels?

Solution
We are looking for the loudness in decibels, so we use the first form of the
decibel scale formula:

The ratio in parentheses is 100, because we are given that the sound is 100 times
as intense as the softest audible sound. We find:

Loudness in dB = 10 log10 100 = 10 × 2 = 20 dB

A sound that is 100 times as intense as the softest possible sound has a loudness
of 20 dB, which is equivalent to a whisper (Table 5.3).

Example 2
How does the intensity of a 57 dB sound compare to that of a 23 dB sound?

Solution
We can compare the loudness of two sounds by working with the second form of
the decibel scale formula. You should confirm for yourself that by dividing the
intensity of Sound 1 by the intensity of Sound 2, we find:

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 Substituting 57 dB for Sound 1 and 23 dB for Sound 2, we have:

Intensity of Sound 1 = 10[(loudness of Sound 1 in dB) - (loudness of Sound 2 in dB)]/10

Intensity of Sound 2

= 10 57-23/10 = 103.4 ≈ 2512

A sound of 57 dB is about 2500 times as intense as a sound of 23 dB.

Inverse Square Law for Sound

Sounds become weaker as the distance from the source increases. For instance,
at an outdoor concert, sitting right in front of the speakers can be almost deafening,
while people a mile away may not hear the music at all. This phenomenon is easy to
understand. Figure 5.16 illustrates the concept. As sound travels from the speaker, it
spreads over an increasingly larger area. This area grows with the square of the
distance from the speaker. For example, at 2 meters, the sound is spread over an area
that is 22 = 4 times larger than the area at 1 meter; at 3 meters, it is spread over an
area 32 = 9 times larger than the area at 1 meter, and so on. Therefore, the intensity
of the sound decreases with the square of the distance. This relationship is known as
the inverse square law, which states that the intensity of sound diminishes
proportionally to the square of the distance from the source. In simpler terms, the
intensity of a sound at a distance d from its source is proportional to 1/𝑑2. This
relationship, known as the inverse square law, applies to many other quantities as
well, including the brightness of light and the strength of gravity.

Figure 5.16 Intensity of sound decreases with the square of the distance
from the source

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 5.4.3 pH Scale for Acidity
Many household products, such as cleansers, drain openers, and shampoo are
labelled for a quantity called pH on their product labels. Chemists use pH to classify
substances as neutral, acidic, or basic (also known as alkaline). Here is how it works:
 Pure water is considered neutral and has a pH of 7.
 Substances with a pH lower than 7 are classified as acids.
 Substances with a pH higher than 7 are classified as bases.
Acidity is chemically related to the concentration of positively charged hydrogen
ions (H+), which are hydrogen atoms without their electrons. The concentration of
hydrogen ions is denoted [H+], typically measured in moles per liter. A mole is a
specific number of particles, with its value defined as 1 mole equal to approximately
6 × 1023 particles, known as Avogadro’s number.
Table 4 Typical pH scale
Solution pH Solution pH
Lemon Juice 2 Pure Water 7
Stomach Acid 2-3 Baking Soda 8.4
Vinegar 3 Household Ammonia 10
Drinking Water 6.5 Drain opener 10-12

pH scale
The pH scale can be defined by the following formulas:
pH = - log10 [H+] or [H+] = 10-pH

Where [H+] is the hydrogen ion concentration in moles per liter. Pure water is neutral
and has a pH of 7. Acids have a pH lower than 7 and bases have a pH higher than 7.

Example
What is the pH of a solution with a hydrogen ion concentration of 10-12 mole per liter?
Is it an acid or a base?

Solution
Using the first version of the pH formula with a hydrogen ion concentration
of [H+] = 10-12 mole per liter, we find:
pH = -log10 [H+] = - log10 10-12 = - (-12) = 12

(Recall that log1010x = x) A solution with a hydrogen ion concentration of 10-12 mole
per liter has pH 12. Because this pH is well above 7, the solution is a strong base.

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Acid Rain
Normal raindrops are slightly acidic, with a pH just below 6. However, the burning of
fossil fuels releases sulfur and nitrogen compounds, which can form sulfuric and nitric
acids in the atmosphere. These acids can make raindrops much more acidic than usual,
leading to a phenomenon known as acid rain. Acid rain has been observed with pH levels
as low as 2 in the northeastern United States and acid fog in the Los Angeles area—
equivalent to the acidity of pure lemon juice.
Acid rain can cause significant harm to the environment. It can kill trees and other plants,
severely damaging forests. Many forests in the northeastern United States and southeastern
Canada have been affected by acid rain. Additionally, acid rain can "kill" lakes by making
the water so acidic that it becomes uninhabitable for aquatic life. Thousands of lakes in
these regions have suffered from acidification. Interestingly, a "dead" lake can often be
recognized by its unusually clear water, as it lacks the living organisms that typically cause
murkiness.

Example
In terms of hydrogen ion concentration, compare acid rain with a pH of 2 to ordinary
rain with a pH of 6.

Solution
For acid rain with pH 2, the hydrogen ion concentration is:

[H+] = 10-pH = 10-2 mole per liter

For ordinary rain with pH 6, the hydrogen ion concentration is:

[H+] = 10-pH = 10-6 mole per liter

Therefore, the hydrogen ion concentration in the acid rain is greater than that in ordinary
rain by a factor of:

10-2 = 10-2-(-6) = 104

10-6

Acid rain is 10,000 times as acidic as ordinary rain.

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Summary
In this unit, we covered linear and exponential growth, learning about how things grow
steadily in a straight line (linear) or rapidly increase over time (exponential). We also
discussed doubling time (how long it takes for something to double in size) and half-life
(how long it takes for something to decrease by half). We examined population growth, the
factors that influence it, theoretical models that predict how populations grow and carrying
capacity, which is the maximum number of people or organisms that an environment can
support. Carrying capacity depends on many factors, such as resource use, environmental
impact, and technological changes. Logarithmic scales are used to measure things that
change exponentially. The magnitude scale for earthquakes, which shows how much
energy an earthquake releases. The decibel scale for sound, which measures how loud a
sound is. The pH scale for acids and bases, which tells us how acidic or basic a substance
is. These scales help us understand and compare the intensity of different phenomena, such
as earthquakes, sounds, and the acidity of substances. Overall, this unit provided a clear
explanation of growth patterns, population dynamics, and measurement scales, helping us
understand how to analyze and interpret various complex systems in our world.

Multiple Choice Questions (MCQs)

1. What is the primary difference between linear and exponential growth?
a) Linear growth increases by multiplication, and exponential by addition.
b) Linear growth increases by addition, and exponential by multiplication.
c) Both grow at the same rate.
d) Exponential growth only occurs in population studies.
Answer: b) Linear growth increases by addition, exponential by multiplication.
2. If a population doubles every 10 years, what type of growth is it exhibiting?
a) Linear
b) Exponential
c) Logarithmic
d) Static
Answer: b) Exponential
3. Which factor is least likely to directly affect population growth?
a) Birth rate
b) Death rate
c) Carrying capacity of the environment
d) Type of currency used in the region
Answer: d) Type of currency used in the region
4. What does “carrying capacity” refer to in population growth models?
a) The maximum population size an environment can sustain
b) The rate at which a population grows

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 c) The average lifespan of individuals in a population
d) The time it takes for a population to double
Answer: a) The maximum population size an environment can sustain
5. What does the pH scale measure?
a) Earthquake magnitude
b) Sound intensity
c) Acidity or alkalinity
d) Population density
Answer: c) Acidity or alkalinity
6. Which of the following is measured in decibels?
a) Earthquake intensity
b) Sound intensity
c) Acidity levels
d) Population growth rates
Answer: b) Sound intensity
7. Doubling time is associated with which type of growth?
a) Linear
b) Static
c) Logarithmic
d) Exponential
Answer: d) Exponential
8. The theoretical models of population growth include which of the following?
a) Exponential and logarithmic models
b) Exponential and linear models
c) Linear and inverse square models
d) Logarithmic and static models
Answer: b) Exponential and linear models
9. What is the inverse square law primarily used to measure?
a) Population growth rates
b) Earthquake magnitude
c) Sound intensity
d) Carrying capacity
Answer: c) Sound intensity
10. What is a key feature of exponential growth?
a) It remains constant over time.
b) It increases by fixed intervals.
c) It is only observed in laboratory settings.

116
 d) It accelerates as the quantity increases.
Answer: d) It accelerates as the quantity increases.

Short-Answer Questions
1. Define linear growth and provide an example.
2. What is exponential growth, and how does it differ from linear growth?
3. Explain the concept of carrying capacity in population growth.
4. What is the significance of doubling time in exponential growth?
5. How is the pH scale used to measure acidity?
6. How do birth rates and death rates impact population growth?
7. Explain the concept of carrying capacity.
8. Explain how the pH scale is used to classify substances as acidic, neutral, or basic.
9. What is a logarithmic scale, and why is it useful?
10. How is sound intensity measured using the decibel scale?
11. Describe the inverse square law and provide an example.

Subjective Questions
1. Discuss the factors affecting population growth and their implications on carrying
capacity.
2. Define doubling time and half-life. How would you calculate the doubling time for
a population growing at a rate of 3% per year?
3. Compare and contrast linear and exponential growth with examples from real-life
situations.
4. Explain the logarithmic scales used for measuring earthquakes and sound intensity.
Why are logarithmic scales useful in these contexts?
5. How does acid rain impact the environment, and why is understanding the pH scale
important in addressing this issue?
6. Analyze the role of theoretical models in understanding population growth. Provide
examples of how these models are applied in real-world scenarios.
7. What are the consequences of unchecked exponential growth in urban populations?
Discuss in terms of resource depletion and environmental sustainability.
8. Read the following scenarios carefully and try answering the question.
i. If a radioactive substance has a half-life of 10 years, how much of a 100g
sample will remain after 30 years?
ii. A small island has a population of rabbits that doubles every year. Initially,
there were 10 rabbits. If the carrying capacity of the island is 320 rabbits,
sketch a graph showing the population growth over time and indicate when
the growth starts to level off.
iii. An earthquake measuring 6 on the Richter scale is recorded. Later, an
earthquake measuring 8 is recorded. How much more energy did the second
earthquake release compared to the first?
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 iv. A sound at a rock concert measures 120 dB. If the sound intensity is halved,
what will be the new decibel level?
v. A sample of rainwater has a pH of 4. How much more acidic is this rainwater
compared to pure water with a pH of 7?
vi. You are standing 2 meters away from a sound source, and the sound intensity
is 80 dB. What will be the intensity if you move to a distance of 4 meters
from the source?

Activities
1) The data on population growth over 10 years is provided below. Plot this data on a
graph and determine whether the trend represents linear or exponential growth.
i) Year 1: 1,000
ii) Year 2: 1,200
iii)Year 3: 1,440
iv) Year 4: 1,728
v) Year 5: 2,074
vi) Year 6: 2,489
vii) Year 7: 2,987
viii) Year 8: 3,585
ix) Year 9: 4,302
x) Year 10: 5,162
2) Use online tools such as PhET Acid Rain Simulator or similar virtual resources to
check the pH level of various substances.
3) Direct students to apps or online resources like "Decibel Meter" apps to measure
sound intensity in various situations such as quiet room, market, classroom, library,
etc.

REFERENCES

Briggs, B.(2015). Using and understanding Mathematics: A quantitative reasoning

approach. USA: Pearson Education Inc.

Cottrell, S. (2015). Critical Thinking Skills. New York: Palgrave Macmillan Ltd.

Starkey, L. (2004). Critical thinking skills success in 20 minutes a day. New York:
Learning Express, LLC.

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Unit–6

MATHEMATICAL MODELS
FOR UNDERSTANDING
PHENOMENA

Written by: Dr. Mubeshera Tufail

Reviewed by: Dr. Muhammad Tanveer Afzal

119
 INTRODUCTION

In this unit, learners explore key concepts such as functions, independent and dependent
variables, and linear and exponential models. These ideas help us understand how different
factors or quantities are connected and how they change over time. Functions describe how
one variable influences another, while linear models focus on steady changes and
exponential models explain how things grow or shrink at a constant rate. By learning about
these models, students develop the skills to analyze data and recognize patterns in real-
world situations.
These concepts are essential for quantitative reasoning, enabling learners to make
informed decisions, predict trends, and solve problems in various fields. This knowledge
equips them with the tools to approach complex challenges and apply mathematical
thinking to everyday life.

Learning Outcomes
At the successful completion of the unit, the students would be able to:
1. Differentiate between independent and dependent variables in mathematical
contexts.
2. Describe the characteristics of linear functions, including constant rates of change
and straight-line graphs.
3. Analyze real-world scenarios and construct linear mathematical models to
represent relationships between variables.
4. Explain the concept of exponential growth and decay and its application in various
contexts.
5. Utilize exponential functions to model phenomena with constant relative growth
rates or decay rates.
6. Apply mathematical modeling techniques to analyze data, make predictions, and
solve problems in real-world scenarios.
7. Critically evaluate the appropriateness of linear and exponential models for
different types of data and phenomena.

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6.1 Concept of Functions in Mathematical Models
Mathematical models serve as crucial tools in understanding and representing real-
world phenomena by translating relationships between varying quantities into
mathematical functions. Just as architectural models and road maps provide a
simplified, yet insightful view of complex structures or landscapes, mathematical
models offer a way to analyze and predict behaviors in diverse fields such as
engineering, economics, and environmental science.

Role of Functions in Mathematical Models

Functions are fundamental to mathematical models because they describe how one
quantity depends on another. A function defines a relationship where each input is
associated with exactly one output. This relationship can be expressed through
formulas, graphs, or tables. Some models may involve a single, straightforward
function. For example, the formula 𝑓(𝑥) = 𝑚𝑥+𝑏 describes a linear relationship where
changes in 𝑥 predict changes in 𝑓(𝑥). More intricate systems, such as climate models,
require multiple interrelated functions to capture the dynamics of the system. These
models might be represented by systems of equations that need powerful computational
resources to solve.
To design a bridge, engineers use functions to relate wind speed to stress and strain
on materials. These models ensure the bridge can withstand various weather conditions.
Economists use models to understand the relationship between worker productivity and
unemployment, helping to predict economic trends.
6.1.1 Language and Notation of Functions
In the ‘Quantitative Reasoning-I course, we discussed how the balance in a savings
plan changes with the interest rate and monthly deposits. Similarly, in unit 05 of this
course, we investigated how population dynamics relate to growth rate and time. These
connections constitute functions because they precisely articulate how one quantity
evolves concerning another. Now, we will examine the language and notation associated
with functions.
6.1.2 Independent and Dependent Variables
A variable is a trait or characteristics whose value change or vary from subject to
subject. For example, temperature, test scores etc. In mathematical and scientific
contexts, variables are often denoted by letters such as 𝑥, 𝑦 or 𝑧. In research, variables
are crucial as they are the elements that are measured, manipulated, or controlled in an
experiment or study.
Consider modeling the temperature fluctuations over a day using the data in Table
6.5. The initial step involves identifying the two relevant quantities: time and
temperature. Our objective is to express the relationship between these variables as a
function. Variables within a function are termed as such because they exhibit change or
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variation. In this context, temperature represents the dependent variable as it is contingent
upon the time of day. Conversely, time acts as the independent variable since it alters
irrespective of temperature changes. We characterize temperature as varying with respect
to time, noting that each time point corresponds to a unique temperature value.
Table 6.5 Temperature Data for One Day
Time Temperature
6:00 am 20 °C
7:00 am 22 °C
8:00 am 25 °C
9:00 am 28 °C
10:00 am 31 °C
11:00 am 33 °C
12:00 pm 35 °C
1:00 pm 38 °C
2:00 pm 38 °C
3:00 pm 34 °C
4:00 pm 33 °C
5:00 pm 31°C
6:00 pm 30 °C

We can find examples of independent and dependent variables in various fields. Some
examples are given below:
 The distance traveled by a car is dependent on the time it has been traveling and
its speed. [Dependent Variable: Distance traveled; Independent Variable: Time
or speed of the car]
 The height of a plant is dependent on the amount of sunlight it receives.
[Dependent Variable: Height of the plant; Independent Variable: Amount of
sunlight]
 The sales revenue of a company is dependent on its advertising expenditure.
[Dependent Variable: Sales revenue; Independent Variable: Advertising
expenditure]
 Test scores of students are dependent on the amount of time they spend
studying. [Dependent Variable: Test scores; Independent Variable: Study time]
 The rate of a chemical reaction is dependent on the concentration of reactants.
[Dependent Variable: Rate of the reaction; Independent Variable:
Concentration of reactants]
6.1.3 Writing Functions

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 When expressing relationships between variables, we commonly organize them as
ordered pairs, with the independent variable listed first and the dependent variable
second:
(time, temperature)
To denote functions, we employ specific notation. For instance, we often use 𝑥 to
represent the independent variable and 𝑦 to denote the dependent variable. Thus, we write
𝑦 = 𝑓(𝑥), which reads as "y is a function of x," indicating that 𝑦 is related to 𝑥 through
the function 𝑓. In the case of our example with time and temperature from Table 6.5, let
𝑡 represent time and 𝑇 represent temperature. Hence, we write 𝑇 = 𝑓(𝑡) to signify that
temperature varies with time or that temperature is a function of time.
It might be helpful to envision a function as a box featuring two slots: one for input
and the other for output (see Figure 6.17). The independent variable's value can be
inserted into the box through the input slot. The function within the box then "operates"
on this input, yielding one value of the dependent variable as output from the box.

Figure 6.17 Function, independent and dependent variable

Numerous functions describe changes over time. For instance, a child's weight and the
Consumer Price Index both fluctuate over time. However, not all functions revolve
around time. Mortgage payments, for instance, are influenced by the interest rate, as the
payment amount depends on this rate. Similarly, a car's gas mileage is contingent on its
speed, with mileage varying at different speeds.
Functions and Variables
A function describes how a dependent variable changes with respect to one or more
independent variables. When there are only two variables, we may denote their
relationship as an ordered pair with the independent variable first:

(independent variable, dependent variable)

It can be said that the dependent variable is a function of the independent variable. If
x is the independent variable and y is the dependent variable, we write the mathematical
equation for the function as:
y= f (x)

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6.1.4 Representing Functions
There are three fundamental methods for representing a function:
 Data Table: Functions can be represented using a data table, similar to Table
6.5. Tables offer detailed information but may become cumbersome when
dealing with large datasets.
 Graph: Another approach is to depict a function graphically. Graphs are easily
interpreted and present a significant amount of information in a compact visual
form.
 Mathematical Equation: Functions can also be represented by mathematical
equations or formulas. These equations provide a compact representation of the
function's behavior.

6.1.5 Domain and Range

The domain of a function refers to the set of values that are both meaningful and
relevant for the independent variable. On the other hand, the range of a function
comprises the values of the dependent variable that correspond to the domain values.
Before plotting a function's graph, it is essential to determine the appropriate variables
for each axis. While mathematically, each axis extends infinitely in both directions, most
functions only hold significance over a limited region of the coordinate plane. For
instance, in the case of the (time, temperature) function based on Table 6.5, negative time
values are nonsensical. The relevant times for this function are those during which data
were collected—from 6 a.m. to 6 p.m. These meaningful values of time constitute the
domain of the function. Similarly, for this function, the temperatures fall between 6:00
a.m. and 6:00 p.m., with recorded temperatures ranging from 28°C to 38°C. Thus,
temperatures within this range make up the function's range. In general, we can define
the following:
 Domain: The set of meaningful and relevant values for the independent variable.
 Range: The set of values of the dependent variable that correspond to the domain
values.
Plotting the Graph for a Function and Coordinate Plane
Standard method for graphing a function involves utilizing a coordinate plane,
which is constructed by drawing two perpendicular number lines. Each of these
number lines is referred to as an axis (plural: axes). Typically, numbers increase to the
right along the horizontal axis and upward along the vertical axis. The point where
both axes intersect, indicating zero on both number lines, is termed the origin. When
dealing with functions in general, the horizontal axis is denoted as the x-axis, while
the vertical axis is termed the y-axis.

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 In the coordinate plane, points are described by pairs of coordinates, known as
ordered pairs. The x-coordinate specifies the point's horizontal position relative to the
origin. Points to the right of the origin have positive x-coordinates, while those to the
left have negative x-coordinates. Meanwhile, the y-coordinate denotes the point's
vertical position relative to the origin. Points above the origin possess positive y-
coordinates, whereas those below have negative y-coordinates. The location of a
specific point is expressed by listing its x- and y-coordinates within parentheses in the
format (x, y). When dealing with functions, the x-axis always represents the
independent variable, and the y-axis represents the dependent variable.
Figure 6.18 (left graph) displays a coordinate plane with various points identified
by their coordinates, where the origin is represented by the point (0, 0). Additionally,
Figure 18 (right graph) illustrates how the axes divide the coordinate plane into four
quadrants, numbered counterclockwise starting from the upper right.

Figure 6.18 Coordinate plane and plotting the graph for a function

After discussing the domain and range, we can proceed to graph the function. We
utilize the horizontal axis to represent time, denoted as 𝑡 with labels indicating hours after
6:00 a.m. Therefore, 𝑡 = 0 corresponds to the initial measurement at 6:00 a.m., and 𝑡 =
12 corresponds to the final measurement at 6:00 p.m. Meanwhile, the vertical axis is
allocated for temperature, denoted as 𝑇. Figure 6.19 illustrates the range and domain.

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 Figure 6.19 Range and Domain
However, temperature varies continuously throughout the day, and every moment has
a corresponding temperature. To create a more realistic model, we should interpolate
between the data points. Since there were no abrupt temperature changes during the day,
it is reasonable to connect the data points with a smooth curve, as depicted in Figure 6.20.

Figure 6.20 Graphing of Function

The resulting graph serves as a model enabling us to predict the temperature at any
given time of day. For instance, the model suggests that the temperature was
approximately 34°C at 11:30 a.m. (5.5 hours after 6:00 a.m.). It is important to note that
while this prediction may not be exact, we lack data for 11:30 a.m. in Table 6.5 to verify
it. Nevertheless, the prediction appears plausible based on our assumptions in drawing
the smooth curve. This example underscores a crucial lesson applicable to all
mathematical models: The accuracy of a model's predictions is contingent upon the
quality of the data and the underlying assumptions.

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 Creating and utilizing graphs of functions typically involves four steps:

Step 1: Identify the independent and dependent variables of the function.

Step 2: Determine the domain (values of the independent variable) and the range
(values of the dependent variable) of the function. Utilize this information to select the
scale and labels for the axes. Consider zooming in on the region of interest to enhance
the graph's readability.
Step 3: Construct a graph using the provided data. If appropriate, interpolate between
data points to fill in any gaps.
Step 4: Before relying on any predictions generated by the model, critically evaluate
the data and assumptions underlying the model's construction.
By following these steps, you can effectively create and utilize graphs of
functions to analyze relationships and make predictions.

6.2 Linear Modeling

Understanding the fundamental principles of mathematical modeling often begins with
simpler models such as linear models, which are represented by linear functions i.e.,
functions characterized by straight-line graphs.
Linear Functions
Consider measuring the depth of rain accumulating in a rain gauge during a steady rainfall
(see Figure 6.21. After 6 hours, the rain ceases, and we seek to describe how the rain depth
changed over time during the storm. In this scenario, time serves as the independent
variable, while rain depth acts as the dependent variable. Suppose, based on our
observations with the rain gauge, we obtain the rain depth function depicted in Figure 6.21.
Given that the graph forms a straight line, we are dealing with a linear function. Utilizing
this linear function as a model to forecast rain depth at different times constitutes
employing it as a linear model.

Figure 6.21 Graph of a function of rain depth over time

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Rate of Change
According to the graph (figure 6.21), the rain depth increased by 1 inch every hour
during the storm. This indicates that the rate of change of the rain depth with respect to
time was constant at 1 inch per hour, or 1 in/hr. Throughout the entire storm, regardless
of the hour chosen for examination, the rain depth consistently rose by 1 inch. This
highlights a fundamental characteristic of linear functions: they maintain a constant rate
of change, resulting in a straight-line graph.
As the rate of change increases, the steepness of the graph also increases. The small
triangles depicted on the graphs illustrate the slope of each line. The slope is defined as
the vertical rise of the graph over a specific horizontal distance traveled. In other words,
it represents the ratio of the rise over time.
Linear Functions
A linear function has a constant rate of change and a straight-line graph. For all
linear functions,
 The rate of change is equal to the slope of the graph.
 The greater the rate of change, the steeper the slope.
 We can calculate the rate of change by finding the slope between any two points
on the graph (Figure 6.22), as follows. To find the slope of a straight line, we
can look at any two points and divide the change in the dependent variable by
the change in the independent variable.

rate of change = slope = change in dependent variable

change in the independent variable

Figure 6.22 Slope of a straight line

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6.3 Exponential Modeling
Besides linear functions, another prevalent and significant growth pattern is
exponential growth, wherein the relative growth rate remains constant. Here, we read
about exponential functions and their manifold applications in mathematical modeling.

6.3.1 Exponential Functions

Consider a town that commences with a population of 10,000 and experiences a
growth rate of 20% per year. As previously discussed in Unit 5, this population exhibits
exponential growth because it increases by the same relative amount, 20%, annually.
Using the terminology from Chapter 8, the percentage growth rate is denoted as 𝑃%=20%
per year, and the fractional growth rate is 𝑟 = 𝑃/100 = 0.2 per year.
At the outset, the population is 10,000. With a 20% growth rate, the population after
the first year amounts to 20% more than the initial population, or 120%, equivalent to 1.2
times the initial population:
Population after 1 year =10,000 × 1.2 = 12,000
During the subsequent year, the population again increases by 20%, leading us to
calculate the population at the end of the second year by multiplying once more by 1.2:

Population after 2 years = Population after 1 year × 1.2 = 12,000 × 1.2 = 14,400

With another 20% growth in the third year, we can determine the population at the
end of the third year by multiplying again by 1.2:

Population after 3 years = Population after 2 years × 1.2 = 14,400 × 1.2 =17,280

The trend becomes evident: if we let 𝑡 represent the time in years, we find that:

Population after 𝑡 years = Initial population × 1.2 𝑡

For instance, after 𝑡= 25, the population amounts to 10,000×1.225= 953,962

This concept can be extended to any exponentially growing quantity 𝑄, allowing us to
formulate a general exponential function.
An exponential function grows (or decays) by the same relative amount per unit of time,
at a rate described by the fractional growth rate r (the “growth” rate is negative, r 6 0, for
decay). The exponential function has the general form
Q = Q0 × (1 + r)t
where
Q = value of the exponentially growing 1 or decaying2 quantity at time t

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Q0 = initial value of the quantity (at t = 0)
r = fractional growth rate for the quantity (or decay rate if r ˂ 0)
t = time
 The units of time used for t and r must be the same. For example, if the fractional
growth rate is 0.05 per month, then t must also be measured in months.
 Remember that while an exponentially growing quantity has a constant relative
growth rate, its absolute growth rate increases. For example, a population may
increase at a constant 20% per year, but the absolute change in the population
each year is always increasing.
Graphing Exponential Functions
Graphing exponential functions is straightforward, primarily involving plotting
points corresponding to several doubling times (or half-lives in the case of decay). We
initiate the graph at the point (0, 𝑄0) representing the initial value at 𝑡 = 0. For an
exponentially growing quantity, we recognize that the value of 𝑄 doubles after one
doubling time (𝑇double)), quadruples after two doubling times (2𝑇double), octuples after
three doubling times (3𝑇double), and so forth. By connecting these points, we can visualize
a steeply rising curve, as illustrated in Figure 6.23.
Conversely, for an exponentially decaying quantity, the value of 𝑄 decreases to half its
initial value (𝑄0/2) after one half-life (𝑇half), one-fourth its initial value (𝑄0/4) after two
half-lives (2𝑇half ), one-eighth its initial value (𝑄0/8) after three half-lives (3𝑇half), and
so on. Plotting these points yields a falling curve, as depicted in Figure 6.23. It is
noteworthy that this curve progressively approaches the horizontal axis but never
intersects it.

Figure 6.23 Exponential growth (b) Exponential decay

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 Summary
In this unit, we learned about functions, independent and dependent variables, and two
types of mathematical models: linear and exponential. Functions help us understand how
different quantities are connected. Independent variables affect the dependent variables.
Linear functions show steady changes and create straight-line graphs, which are good for
modeling situations with consistent growth or decline. On the other hand, exponential
functions show how things grow or shrink at a constant rate, leading to faster or slower
changes over time. By understanding these concepts, we can use them to solve real-world
problems, predict future trends, and make better decisions.

Multiple Choice Questions (MCQs)

1) Which of the following best defines a function in mathematics?
a) A process that maps multiple outputs to a single input.
b) A relation where each input corresponds to exactly one output.
c) A rule that maps outputs randomly.
d) A formula used in solving equations.
Answer: b) A relation where each input corresponds to exactly one output.
2) Which of the following statements correctly describes the relationship in a function?
a) A function can have one input corresponding to multiple outputs.
b) A function always has multiple inputs for a single output.
c) Each input in a function corresponds to exactly one output.
d) Functions have no restrictions on input-output relationships.
Answer: c) Each input in a function corresponds to exactly one output.
3) Why is it important to identify the type of relationship in a function?

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 a) To solve mathematical puzzles.
b) To ensure the function is invertible.
c) To apply the function correctly in real-world scenarios.
d) To calculate values for undefined variables.
Answer: c) To apply the function correctly in real-world scenarios
4) Which of the following is a real-world application of a linear function?
a) Measuring the depreciation of a car's value over time.
b) Modeling the growth of bacteria in a lab.
c) Analyzing the spread of a virus in a population.
d) Estimating population growth with doubling time.
Answer: a) Measuring the depreciation of a car's value over time.
5) What is the domain of a function?
a) The range of possible outputs.
b) The set of all possible inputs for which the function is defined.
c) The difference between the highest and lowest values of the function.
d) The area under the graph of the function.
Answer: b) The set of all possible inputs for which the function is defined.
6) Exponential models are most commonly used to represent:
a) Relationships that decrease linearly over time.
b) Growth processes that increase rapidly as time progresses.
c) Fixed and unchanging relationships.
d) Constant rates of change.
Answer: b) Growth processes that increase rapidly as time progresses.
7) Why is it important to understand the domain and range of a function?
a) To determine the graph's color.
b) To understand the function’s applicability and limitations.
c) To avoid solving the function.
d) To identify the dependent variable only.
Answer: b) To understand the function’s applicability and limitations.
8) What distinguishes a linear function from an exponential function?
a) A linear function has a constant rate of change, while an exponential function
does not.
b) An exponential function is always a straight line.
c) A linear function has a variable rate of change.
d) Both have the same rate of change.

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 Answer: a) A linear function has a constant rate of change, while an exponential
function does not.
9) What is a practical example of an exponential model in real life?
a) The relationship between hours worked and wages earned.
b) Population growth under ideal conditions.
c) Speed of a car traveling at a constant rate.
d) Monthly expenses in a fixed budget.
Answer: b) Population growth under ideal conditions.
10) How does mathematical modeling help in understanding phenomena?
a) It guarantees absolute accuracy in predictions.
b) It avoids the need for understanding real-world variables.
c) It eliminates variability in observed data.
d) It simplifies complex systems into manageable representations.
Answer: d) It simplifies complex systems into manageable representations.
Short Answer Questions
1. Define a function and explain its importance in mathematical modeling.
2. What are independent and dependent variables? Provide an example.
3. Describe the difference between the domain and range of a function.
4. How can you identify whether a situation represents linear or exponential growth?
5. Why is it essential to understand the type of function before applying it to real-
world scenarios?
Subjective Questions
1. Explain the difference between linear and exponential functions with examples.
Discuss their definitions, characteristics, and applications in real-life situations.
2. Discuss the role of mathematical models in understanding and solving real-world
problems. Highlight how functions, domain, and range contribute to accurate
modeling.
3. Describe the process of constructing a mathematical function for a real-world
scenario. Include steps like identifying variables, determining the relationship, and
writing the function.
4. Compare and contrast linear and exponential models in terms of their applicability
and limitations. Use examples such as population growth and salary progression to
support your answer.
5. Explain the importance of understanding independent and dependent variables in
creating mathematical models. Give examples from everyday life, such as speed and
travel time.

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6. A small town had a population of 5,000 in the year 2000. Since then, the town's
population has been growing at a constant rate of 3% per year. Create a linear and
an exponential model for the town's population growth. Use these models to predict
the population in the year 2025. Discuss which model (linear or exponential) is more
appropriate and why.
7. Consider the temperature data recorded every hour from 6:00 a.m. to 6:00 p.m. in a
particular city. The data shows a steady increase in temperature until noon, followed
by a gradual decline. Identify the independent and dependent variables in this
scenario. Create a graph to represent the temperature variation throughout the day.
Determine whether a linear or exponential model best fits the data and justify your
choice.
8. During a steady rainstorm, a rain gauge shows the following depths of accumulated
rain: 0.5 inches after the first hour, 1 inch after the second hour, 1.5 inches after the
third hour, and so on. Plot the data on a graph and determine if the relationship
between time and rain depth is linear. Derive the linear function representing this
relationship and use it to predict the rain depth after 10 hours.
9. A certain radioactive substance decays at a rate of 5% per year. Initially, there are
200 grams of the substance. Develop an exponential model for the decay of this
substance. Use the model to calculate the amount of the substance remaining after
10 years. Explain why an exponential model is more suitable for this scenario than
a linear model.
10. An initial investment of Rs. 1,000/- is made in an account that offers an annual
interest rate of 8%, compounded yearly. Construct an exponential model to represent
the growth of this investment over time. Determine the value of the investment after
5 years and after 20 years. Discuss the advantages of using an exponential model for
financial growth predictions.
11. A biologist is observing the growth of a bacterial culture that doubles in number
every 3 hours. Initially, there are 1,000 bacteria. Create a table and graph showing
the number of bacteria at 0, 3, 6, 9, and 12 hours. Formulate an exponential function
to model the bacterial growth and use it to predict the number of bacteria after 24
hours. Explain the reasoning behind using an exponential function for this type of
growth.
12. The value of a new car depreciates by 15% each year. If the initial value of the car
is Rs. 3,000,000/- develop a model to represent the car's depreciation over time. Use
the exponential decay model to estimate the value of the car after 5 years. Discuss
the implications of using an exponential model for depreciation as opposed to a
linear model.

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 13. An infectious disease spreads through a population such that the number of infected
individuals doubles every 4 days. Initially, there were 50 infected individuals.
Construct an exponential model to represent the spread of the disease. Predict the
number of infected individuals after 2 weeks. Analyze the effectiveness of using an
exponential model in this context and discuss any limitations.

Activities
1) List three examples from daily life where one thing depends on another (e.g., the time
it takes to cook rice depends on the heat level). Write what changes (dependent
variable) and what causes the change (independent variable).
2) Match the following situations to their type of function (linear or exponential). Write
"linear" or "exponential" for each example and explain why.
 A car’s speed increases by 10 km/h every minute. (Change in the speed after
30 minutes)
 The population of a city doubles every 10 years. (change in population after
50 years)
3) Choose one situation. Draw a simple graph showing how earnings or savings change
over time.
 A worker earns Rs.150/- for each hour worked. (30 hours worked in one week)
 A savings account grows by 5% every year. (profit earned in next 10 years)
4) For the following scenarios, identify the domain (possible input values) and range
(possible output values):
 A student scores between 0 and 100 on an exam.
 The height of a ball thrown into the air changes over time (from 0 seconds to
5 seconds).
5) Create your own example of a situation that can be modeled with a function. Describe
the situation (e.g., buying apples at Rs.300/- per kg) and write a function for it (e.g.,
Cost = 300 × Weight).

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 REFERENCES

Briggs, B.(2015). Using and understanding Mathematics: A quantitative reasoning

approach. USA: Pearson Education Inc.

Cottrell, S. (2015). Critical Thinking Skills. New York: Palgrave Macmillan Ltd.

Starkey, L. (2004). Critical thinking skills success in 20 minutes a day. New York:
Learning Express, LLC.

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Unit–7

CORRELATION AND CAUSALITY

Written by: Dr. Mubeshera Tufail

Reviewed by: Dr. Shahbaz Hamid

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 INTRODUCTION

Quantitative reasoning involves solving problems using numbers and patterns, which
are important when studying relationships between things. This unit is connected to
quantitative reasoning because it helps us understand and analyze data better.
In this unit, you will learn about correlation and causality. Correlation means the
relationship between two variables. Causality helps to understand whether the change in
one variable causes the change in another variable. Understanding these concepts will be
helpful to make sense of data and research in many areas, like science and everyday life. It
would also improve your ability to think critically and draw correct conclusions.

Learning Outcomes
After the successful completion of the unit, the students would be able to:
1. Describe the concept of correlation and how it shows the link between two
variables.
2. Explain the difference between positive, negative, and no correlation.
3. Read scatterplots to find the type and strength of a correlation.
4. Understand the difference between correlation and causality.
5. Adopt the steps to check causality, like controlling for other factors and using
experiments to find cause-effect relationships.
6. Study examples of correlation and causality in areas like health, social studies, and
economics.

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7.1 Correlation
Correlation is a statistical measure that describes the extent to which two or more
variables fluctuate together. A positive correlation indicates that as one variable increases,
the other tends to increase as well, while a negative correlation indicates that as one variable
increases, the other tends to decrease. The strength and direction of a correlation are
typically quantified using a correlation coefficient, often denoted by 𝑟, which ranges from
-1 to 1.
 Positive Correlation (0 < 𝑟 ≤ 1): Indicates that the variables tend to increase or
decrease together.
 Negative Correlation (−1 ≤ 𝑟 < 0): Indicates that one variable tends to increase
when the other decreases.
 Zero Correlation (𝑟 = 0): Indicates no linear relationship between the variables.

Example 1: Studying the relationship between hours studied and exam scores among
students.
In this case, there are two variables ‘Hours studied’ and ‘Exam scores’. An increase
in the number of hours studied causes an increase in the exam scores. It suggests a
positive correlation. For instance, a correlation coefficient (𝑟) of 0.85 would indicate
a strong positive correlation, meaning students who study more tend to score higher
on exams.

Example 2: Investigating the relationship between stress levels and quality of sleep.
In this example, there are two variables: stress levels and quality of sleep. As stress
levels increase, the quality of sleep tends to decrease. It suggests a negative
correlation. For example, a correlation coefficient (𝑟) of -0.65 would indicate a
moderate negative correlation, meaning higher stress is associated with poorer sleep
quality.

Example 3: There is a correlation between the variables height and weight for
people. That is, taller people tend to weigh more than shorter people.

Example 4: There is a correlation between the variables of demand for apples and
the price of apples. That is, demand tends to decrease as prices increase.

Example 5: There is a correlation between practice time and skill among piano
players. That is, those who practice more tend to be more skilled.

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 It is important to remember that correlation does not imply causation. Just because two
variables are correlated does not mean that one causes the other to occur. Other underlying
factors or variables may influence both correlated variables, leading to a spurious
correlation. Always remember:
1. The correlation may be a coincidence.
2. Both variables might be directly influenced by some common underlying cause.
3. One of the correlated variables may actually be a cause of the other (further analysis
may confirm or deny it). Note that, even in this case, it may be only one of several
causes.
Correlation is a fundamental concept in statistics and research, providing insights into the
relationships between variables. By understanding and interpreting correlations,
researchers can identify patterns, make predictions, and formulate hypotheses for further
investigation. However, it is crucial to distinguish between correlation and causation to
avoid incorrect conclusions about the nature of these relationships.
Scatterplots
Remember the types of gaps we studied study in Quantitative Reasoning-I. A scatterplot is
a graph in which each point represents the values of two variables Scatterplot can be used
for showing the relationship between the variables.
In Figure 7.24, the scatterplot for stories shows dots scattered without any discernible
pattern, indicating little to no correlation between the amount of money spent on writing
the story and the amount of money earned in gross receipts. This lack of a clear trend
suggests that higher production costs do not necessarily lead to higher earnings, implying
a very weak or non-existent linear relationship between these two variables (cost and
earning on the story-writing). Conversely, the scatterplot of diamonds in Figure 7.25 shows
a clear upward trend, illustrating a positive correlation between the weights (in carats) and
the retail prices of diamonds. Despite the correlation not being perfect, since the heaviest
diamond is not the most expensive, the general trend is evident: larger diamonds tend to
have higher prices. This positive correlation highlights a more predictable relationship
where increases in weight generally correspond to increases in price. Thus, while the story-
writing data suggests a complex or non-linear relationship between production costs and
earnings, the diamond data clearly supports the intuitive idea that bigger diamonds cost
more.
Table 7.6 Cost and Profit in Story-writing Project
Story-writing Production Cost (millions of Gross Receipt (millions of
Projects dollars) dollars)
Project 1 300 66
Project 2 300 309
Project 3 270 305
Project 4 260 201

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Project 5 258 337
Project 6 250 302
Project 7 250 241
Project 8 250 448
Project 9 237 761
Project 10 232 200
Project 11 230 169
Project 12 225 142
Project 13 225 423
Project 14 220 623
Project 15 215 180

Figure 7.24 Scatterplot for stor-writing project cost and profit

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 Figure 7.25 Scatterplot of price and weight of diamond
Figure 7.26 shows a scatterplot for two variables: life expectancy and child
mortality in 15 countries. It is evident that there is a clear trend for a negative
correlation between the two variables. Countries with higher life expectancy tend to
have lower infant mortality.
Besides stating whether a correlation exists, we can also see its strength. The more
closely the data follow the general trend, the stronger is the correlation.

Figure 7.26 Scatterplot of life expectancy and child mortality

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 Relationships Between two Data variables
 Positive correlation: Both variables tend to increase (or decrease) together.
 Negative correlation: The two variables tend to change in opposite directions,
with one increasing while the other decreases.
 No correlation: There is no apparent relationship between the two variables.
 Strength of a correlation: The more closely two variables follow the general
trend, the stronger the correlation (which may be either positive or negative). In
a perfect correlation, all data points lie on a straight line.

7.2 Causality
When a correlation is observed and causality is suspected, testing this suspicion
involves a rigorous examination of multiple lines of evidence. For this purpose, you have
to recall the concept of dependent and independent variables. Let us discuss how the
causality between smoking and lung cancer was established. As there were ethical
constraints against conducting controlled experiments on humans it was not possible to
conduct an experiment on human beings.
1. Multiple Groups: Researchers observed correlations between smoking and lung
cancer across diverse populations, including different genders, races, and cultures.
This consistency across varied groups strengthens the case for a causal relationship.
2. Comparative Incidence: Among groups of people with similar characteristics,
nonsmokers had significantly lower rates of lung cancer compared to smokers. This
comparison helps to isolate smoking as a distinguishing factor.
3. Dose-Response Relationship: Higher rates of lung cancer were found among those
who smoked more cigarettes and for longer durations. This dose-response
relationship suggests that increased exposure to smoking increases the risk of lung
cancer, which is a key indicator of causality.
4. Controlling for Confounding (i.e. other) Variables: Researchers accounted for
other potential causes of lung cancer, such as radon gas and asbestos exposure.
After controlling for these factors, almost all remaining lung cancer cases were
found among smokers, further implicating smoking as the primary cause.
Additional Lines of Evidence
To address the possibility that an unknown factor, like genetics, predisposes people
to both smoking and lung cancer, further evidence was gathered:
1. Animal Experiments: Controlled experiments on animals involved randomly
assigning them to treatment (exposed to cigarette smoke) and control groups. The
observed correlation between inhalation of cigarette smoke and lung cancer in these
experiments supports the causal link and helps rule out genetic predisposition as
the sole cause.

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 2. Cellular Mechanisms: Biologists studying human lung tissue cultures discovered
how ingredients in cigarette smoke can cause cancerous mutations. This
mechanistic understanding provides a direct link between smoking and the
development of lung cancer, independent of genetic factors.
3. Second-Hand Smoke: The association between exposure to second-hand smoke
and lung cancer in nonsmokers further supports the causality. Since second-hand
smoke affects nonsmokers, it diminishes the likelihood that genetics alone are
responsible and aligns with the understanding that carcinogens in cigarette smoke
cause mutations leading to cancer.
Summary of Guidelines for Establishing Causality
The case for causality is stronger when:
 Correlations are consistent across various populations.
 Comparisons show lower incidence rates in unexposed groups.
 A dose-response relationship is observed.
 Confounding (i.e., other) variables are controlled, and the correlation persists.
 Experimental evidence from animals supports the correlation.
 Biological mechanisms explaining the process are discovered.
 Indirect evidence, such as effects on nonsmokers, aligns with the causal hypothesis.
By meeting these guidelines, researchers can build a robust case for causality, as
demonstrated in the relationship between smoking and lung cancer.
Guidelines for establishing Causality
If you suspect that a particular variable (the suspected cause or independent variable)
is causing some effect in the dependent variable (we also call it a cause-effect
relationship):
1. Look for situations in which the effect (dependent variable) is correlated with
the suspected cause (independent variable) even while other factors vary.
2. Among groups that differ only in the presence or absence of the suspected cause,
check that the effect is similarly present or absent.
3. Look for evidence that larger amounts of the suspected cause produce larger
amounts of the effect.
4. If the effect might be produced by other potential causes (besides the suspected
cause), make sure that the effect still remains after accounting for these other
potential causes.
5. If possible, test the suspected cause with an experiment. If the experiment cannot
be performed with humans for ethical reasons, consider doing the experiment
with animals, cell cultures, or computer models.
6. Try to determine the physical mechanism (for example, experiment on animals
in the example of smoking and lung cancer) by which the suspected cause
produces the effect.

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7.2.1 Confidence in Causality
Determining confidence in a cause-and-effect relationship requires evaluating the
strength of the evidence, much like legal standards of proof. The three broad levels of
confidence are: preponderance of the evidence, where the evidence suggests that something
is more likely true than not; clear and convincing evidence, where the evidence is highly
and substantially likely to be true; and beyond a reasonable doubt, where the evidence is
so strong that there is no reasonable doubt of its truth. For instance, the link between
smoking and lung cancer progressed from early studies suggesting a probable connection
(preponderance of the evidence) to a substantial body of consistent and high-quality
research (clear and convincing evidence), and finally to overwhelming evidence from
multiple disciplines that left virtually no room for doubt (beyond a reasonable doubt). By
aligning research evidence with these standards, we can gauge our confidence in causality
and make informed decisions even when the case is not fully established.
Broad Levels of Confidence in Causality
1 Possible cause: We have discovered a correlation, but cannot yet determine whether
the correlation implies causality. In the legal system, possible cause (such as thinking
that a particular suspect possibly caused a particular crime) is often the reason for
starting an investigation.
2 Probable cause: We have good reason to suspect that the correlation involves cause,
perhaps because some of the guidelines for establishing causality are satisfied. In the
legal system, probable cause is the general standard for getting a judge to grant a
warrant for a search or wiretap.
3 Cause beyond reasonable doubt: We have found a physical model that is so
successful in explaining how one thing causes another that it seems unreasonable to
doubt the causality. In the legal system, cause beyond a reasonable doubt is the usual
standard for conviction. It generally demands that the prosecution show how and
why (essentially the physical model) the suspect committed the crime. Note that
beyond reasonable doubt does not mean beyond all doubt.

Summary
This unit has provided a comprehensive overview of the basic concepts of correlation
and causality, essential for understanding relationships between variables in research and
data analysis. Correlation describes the degree and direction of the relationship between
two variables. A positive correlation indicates that two variables tend to increase together
and vice versa. A negative correlation suggests that as one variable increases, the other
tends to decrease. Zero correlation signifies no linear relationship between the variables.
Through the interpretation of scatterplots, we learned how to visually identify the type and
strength of correlation between variables. Scatterplots serve as powerful tools for
visualizing data and assessing the degree of association between variables.

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 A crucial distinction was made between correlation and causality. While correlation
indicates a relationship between variables, causality implies that changes in one variable
directly cause changes in another. Understanding this difference is essential for avoiding
misinterpretation of data and drawing valid conclusions from research findings. The
criteria and methods used to assess causality, including controlling for confounding
variables and conducting experiments to establish cause-and-effect relationships were
covered in this unit. Overall, this unit has equipped us with the foundational knowledge
and analytical skills necessary to critically evaluate correlations and causality in scientific
studies.

Multiple-Choice Questions
1. What does a correlation coefficient measure?
a) The cause of a relationship between two variables
b) The direction and strength of a relationship between two variables
c) The mean and median of the two datasets
d) The frequency of occurrences in a dataset
Answer: b) The direction and strength of a relationship between two variables
2. Which of the following values indicates the strongest correlation?
a) -0.2
b) 0.5
c) -0.9
d) 0.1
Answer: c) -0.9 (Correlation value, if close to +1/-1 depicts a stronger correlation
between two variables.)
3. What does a correlation coefficient of 0 indicate?
a) A strong relationship between variables
b) A weak positive relationship
c) No relationship between variables
d) A direct causal relationship
Answer: c) No relationship between variables
4. A scatterplot with points forming an upward slope indicates:
a) Negative correlation
b) Positive correlation
c) No correlation
d) Causality
Answer: b) Positive correlation

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5. Which of the following best represents a negative correlation?
a) Hours studied and grades earned
b) Temperature and ice cream sales
c) Price of a product and quantity sold
d) Time spent exercising and weight gained
Answer: c) Price of a product and quantity sold
6. Causality can be confidently established when:
a) Two variables show a strong correlation
b) Experiments eliminate alternative explanations
c) Scatterplots indicate linear relationships
d) The correlation coefficient equals 1
Answer: b) Experiments eliminate alternative explanations
7. Which graph is typically used to visualize the strength and direction of a
correlation?
a) Scatterplot
b) Bar graph
c) Pie chart
d) Line graph
Answer: a) Scatterplot
8. What is a possible reason for a strong correlation that does not imply causation?
a) Experimental evidence
b) A third variable influencing both factors
c) Lack of sufficient data
d) An inverse relationship
Answer: b) A third variable influencing both factors
9. If two variables have no systematic pattern in a scatterplot, their correlation is
likely to be:
a) Positive
b) Negative
c) Neutral
d) Unknown
Answer: c) Neutral
10. What does it mean if a scatterplot shows a tight cluster of points forming a
straight line?
a) The relationship is non-linear.
b) The correlation is weak.
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 c) The correlation is strong.
d) There is no correlation.
Answer: c) The correlation is strong.
Short-Answer Questions
1) Define correlation and causality. How are they different?
2) What is the significance of the correlation coefficient in interpreting data
relationships?
3) How does a scatterplot help in understanding the strength of a correlation?
4) Explain the concept of a positive and negative correlation with examples.
5) Why does correlation not always imply causation?
Subjective Questions
1) Discuss the importance of understanding correlation in real-world scenarios with the
help of examples such as studying the relationship between exercise and health.
2) Explain how causality can be confidently established in research. Mention
experimental methods, control variables, and confounding factors.
3) Describe the steps to analyze a scatterplot and interpret its meaning. Include how to
identify trends, clusters, and outliers.
4) What are some real-life implications of misinterpreting correlation as causation?
Provide examples such as advertising claims or public health policies.
5) How can the strength and direction of a correlation impact decision-making? Use
examples from education, economics, or healthcare.
6) What type of correlation would you expect to find between hours of sleep and the
grades in the study? Why?
7) If you were to plot the data on a scatterplot, what would it look like if there was a
strong positive correlation between hours of sleep and grades in the study? How
would you interpret a scatterplot showing a weak or no correlation between the two
variables?
8) Explain why finding a correlation between hours of sleep and grades in the study
does not necessarily imply that one variable causes changes in the other. What
additional evidence or research design would be needed to establish causality in this
scenario?
9) Describe the criteria you would use to assess causality between hours of sleep and
grades in the previous class in your study. How would you control for potential
confounding variables that could influence GPA, such as study habits or course load?
10) Suppose you found a correlation between hours of sleep and grades in the study, and
controlled for confounding variables in your analysis. What additional evidence
would strengthen your case for causality? How would you incorporate multiple lines

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 of evidence, such as experimental studies or biological mechanisms, to support your
hypothesis of a causal relationship?
Activities
1) Draw a scatterplot for the following data and identify whether the correlation is
positive, negative, or neutral.
Hours Studied Test Scores
1 50
2 60
3 70
4 80
5 90

2) Find a news article that mentions a correlation. Write a short summary and discuss
if the correlation implies causation.
3) For the statement "Students who eat breakfast tend to perform better in school," list
potential other variables that might influence this relationship.
4) Match correlation coefficients to their descriptions.
Correlation Coefficient Description
-0.8 Strong negative
0 No correlation
0.3 Weak positive
0.9 Strong positive

5) Compile three examples of positive, negative, and neutral correlations from your
daily life with a 1-2 sentence explanation for each.

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 REFERENCES

Briggs, B.(2015). Using and understanding Mathematics: A quantitative reasoning

approach. USA: Pearson Education Inc.

Cottrell, S. (2015). Critical Thinking Skills. New York: Palgrave Macmillan Ltd.

Starkey, L. (2004). Critical thinking skills success in 20 minutes a day. New York:
Learning Express, LLC.

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Unit–8

HYPOTHESIS TESTING

Written by: Dr. Mubeshera Tufail

Reviewed by: Dr. Muhammad Rafiq

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 INTRODUCTION

A hypothesis is a prediction or assumption about something that can be tested. It is a

starting point for research that helps to ask questions and find answers. Hypotheses guide
the researchers in understanding patterns and relationships in data.
Hypothesis testing is a method used to check if a hypothesis is true or not. Researchers
collect sample data, analyze it, and decide whether to accept or reject their hypothesis. It is
a useful tool for making decisions based on the information collected from the sample. In
research, hypothesis testing is used in many areas, like studying new medicines, checking
the effects of programs, or comparing groups. The unit will cover the risks in hypothesis
testing like Type I and Type II errors, and the ways to reduce them.
The hypothesis testing process is closely linked to quantitative reasoning. Hypothesis
testing involves analyzing numbers, interpreting results, and making decisions. It teaches
us to think critically and solve problems using data, making it an essential skill for research
and analysis.

Learning Outcomes
After the successful completion of this unit, the students would be able to:
1. Understand the concept of hypothesis and the difference between null and
alternative hypotheses.
2. Explain the significance of hypotheses in scientific inquiry and research
design.
3. Understand the procedure of hypothesis testing using significance levels.
4. Formulate hypotheses for different research questions and scenarios.
5. Make informed decisions about rejecting or failing to reject the null
hypothesis based on statistical evidence.
6. Identify and minimize the risks associated with Type I and Type II errors in
hypothesis testing.
7. Apply hypothesis-testing techniques to analyze real-world data and draw
meaningful conclusions.

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8.1 Defining hypothesis
A hypothesis (singular: hypothesis & plural: hypotheses) is a proposed explanation or
prediction about a phenomenon that can be tested through research and experimentation. It
is an educated guess based on prior knowledge and observations, formulated to be tested
and either supported or refuted. In scientific research, a hypothesis typically specifies the
relationship between two or more variables and is structured in a way that allows for
empirical testing and analysis.
8.1.1 Significance of Hypothesis
Hypotheses play a crucial role in guiding the research process, informing
experimental design, and facilitating the interpretation of results. The significance of
the hypothesis is explained below.
i. Guiding Research Questions: Hypotheses provide a framework for posing specific
research questions and formulating testable predictions. They help researchers to
focus their investigations and clarify the objectives of their studies.
ii. Formulating Predictions: Hypotheses allow researchers to make predictions about
the outcomes of their experiments or observations. By specifying the expected
relationships between variables or the expected characteristics of populations,
hypotheses provide a basis for evaluating the evidence collected during the study.
iii. Testing Theoretical Assumptions: Hypotheses allow researchers to test theoretical
assumptions and hypotheses derived from existing knowledge or theoretical
frameworks. Through hypothesis testing, researchers can evaluate the validity of
these assumptions and contribute to the refinement and development of theories.
iv. Driving Experimental Design: Hypotheses guide the design of experiments by
specifying the variables to be measured and the conditions under which observations
will be made. It ensures that experiments are structured to test specific hypotheses
and yield meaningful results.
v. Interpreting Results: Hypotheses provide a basis for interpreting the results of
research studies. By comparing observed data to the predictions specified by the
hypotheses, researchers can assess the strength of evidence for or against their
hypotheses and draw conclusions about the phenomena under investigation.
vi. Making Informed Decisions: Hypotheses help researchers make informed
decisions about the validity of their findings and the implications of their research.
By evaluating the evidence in light of the hypotheses, researchers can determine
whether their results support or refuse their initial predictions.
vii. Advancing Knowledge: Hypotheses drive the advancement of knowledge by
generating new insights and discoveries. Whether hypotheses are confirmed or
rejected, the process of testing them helps to deeply understand the underlying
phenomena and contributes to the accumulation of scientific knowledge.

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 In summary, hypotheses are significant in research and scientific inquiry because
they provide a framework for generating testable predictions, guiding experimental
design, interpreting results, and advancing knowledge in various fields of study. They
serve as the cornerstone of the scientific method, facilitating the systematic
investigation of hypotheses and the generation of empirical evidence to support or
refute them.
8.1.2 Null and Alternative Hypotheses
Hypothesis testing involves formulating two hypotheses: the null hypothesis,
denoted by 𝐻0 (pronounced as H naught), and the alternative hypothesis, denoted
by 𝐻1 or sometimes 𝐻𝑎. The null hypothesis represents the assumed fact that serves
as a baseline for comparison with a sample. It is typically expressed in terms of
equality. For example, when tossing a coin the null hypothesis would state that the
coin is fair, or in terms of proportions that there is an equal chance for head and tail
or the proportion of heads is 0.5 or 50%. Symbolically, it is written as 𝐻0: 𝑝 = 0.5,
where 𝑝 is the expected proportion of heads for a fair coin.
The alternative hypothesis, also known as the research hypothesis, challenges
the accepted standard by asserting that it is not true. It is generally expressed in
terms of an inequality. For instance, in the context of coin tossing, the alternative
hypothesis might claim that the coin is biased, producing more heads than expected
or the chances of heads being more than tails while tossing a coin or vice versa.
Symbolically, this is represented as 𝐻1: 𝑝> 0.5 indicating that when tossing a coin,
the proportion of heads is greater than 0.5 or 50% i.e., the coin is not fair or the
chances for head and tail are not 50%-50%.
8.1.3 Way of Writing Hypothesis
Some examples are given below to share with you the technique of writing the
hypothesis for a given research question.
Example 1
Research Question: Does caffeine consumption improve reaction time in drivers?
Null Hypothesis (H0): There is no statistically significant difference in reaction time
between drivers who consume caffeine and those who do not.
Alternative Hypothesis (H1): Drivers who consume caffeine will have faster reaction
times compared to those who do not.
Example 2
Research Question: Does exposure to sunlight affect plant growth?
Null Hypothesis (H0): There is no statistically significant difference in plant growth
between plants exposed to sunlight and those kept in the shade.
Alternative Hypothesis (H1): Plants exposed to sunlight will exhibit greater growth
compared to those kept in the shade.

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Example 3
Research Question: Does music tempo influence exercise performance?
Null Hypothesis (H0): There is no statistically significant difference in exercise
performance between individuals who listen to fast-paced music and those who listen
to slow-paced music.
Alternative Hypothesis (H1): Individuals who listen to fast-paced music will
demonstrate better exercise performance compared to those who listen to slow-paced
music.
Example 4
Research Question: Does gender affect mathematical problem-solving abilities?
Null Hypothesis (H0): There is no statistically significant difference in mathematical
problem-solving abilities between males and females.
Alternative Hypothesis (H1): Males will demonstrate better mathematical problem-
solving abilities compared to females.
Example 5
Research Question: Does mindfulness meditation reduce stress levels?
Null Hypothesis (H0): There is no statistically significant difference in the stress levels
between individuals who practice mindfulness meditation and those who do not.
Alternative Hypothesis (H1): Individuals who practice mindfulness meditation will
have lower stress levels compared to those who do not.
Example 6
Research Question: Does online tutoring improve academic performance in high
school students?
Null Hypothesis (H0): There is no statistically significant difference in the academic
performance between high school students who received online tutoring and those who
did not.
Alternative Hypothesis (H1): High school students who receive online tutoring will
demonstrate better academic performance compared to those who do not.
Example 7
Research Question: Does sleep duration affect memory consolidation?
Null Hypothesis (H0): There is no statistically significant difference in memory
consolidation between individuals who sleep for different durations.
Alternative Hypothesis (H1): Individuals who sleep for longer durations will exhibit
better memory consolidation compared to those who sleep for shorter durations.

Example 8
Research Question: Does temperature impact the rate of chemical reactions?
Null Hypothesis (H0): There is no statistically significant difference in the rate of
chemical reactions at different temperatures.

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 Alternative Hypothesis (H1): Chemical reactions will occur at a faster rate at higher
temperatures compared to lower temperatures.
These examples illustrate how hypotheses can be formulated to investigate various
research questions across different fields of study.
8.1.4 Hypotheses Testing
Advertisers assert the benefits of their products. Schools proclaim the quality of
education they provide. Lawyers argue for a suspect's guilt or innocence. Medical
diagnoses involve claims about the presence or absence of disease. Pharmaceutical
companies promote the effectiveness of their drugs. But how can we verify the truth of
these claims? Statistics provides a solution through a powerful set of methods known
as hypothesis testing, which is fundamental to many aspects of modern life.
Imagine a school introduces a new tutoring program and claims it improves
students' math test scores. To test this claim, we can use hypothesis testing.
(i) Hypotheses
Null Hypothesis (𝐻0): The null hypothesis states that the new tutoring program has
no effect on the student's math test scores.
𝐻0: The average test score with the tutoring program is the same as without it.
Alternative Hypothesis (𝐻1): The alternative hypothesis suggests that the new
tutoring program does improve the students' math test scores.
𝐻1: The average test score with the tutoring program is higher than without it.
(ii) Collect Data: Select a group of 100 students and have them take a math test
before starting the tutoring program. After a period of tutoring, have the same
students take a similar math test.
(iii) Calculate Average Scores: Compute the average scores of the tests of the
students from the test conducted in the previous step, for both “before” and
“after” the tutoring program.
(iv) Compare the “before” and “after” test scores: Compare the average scores
of students before and after the tutoring program. (Note: A statistical test i.e.,
paired t-test is used for comparison in this case.)
(v) Results are significant or not: Determine if the difference is statistically
significant (Note: a significance level (e.g., 𝛼=0.05) to decide if the
improvement in scores is statistically significant).
(vi) Outcomes: We can consider rejecting either null or alternative hypotheses
based on the following results. Please note that we can only reject one
hypothesis in this case.
Reject 𝐻0 and fail to reject H1: If the test results show a significant increase in
the average test scores after the tutoring program, we reject the null hypothesis.
This suggests that the tutoring program does improve test scores.

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 Reject H1 and fail to reject 𝐻0: If the test results do not show a significant
increase in the “after” test results as compared to the “before” test result, then
it suggests that there is not enough evidence to conclude that the tutoring
program improves test scores.
This simple example helps to determine if the new tutoring program is effective in
enhancing students' math test scores. In general, the null hypothesis should make a
specific claim about the value of a population parameter, such as a population mean or
proportion. In this case, the null hypothesis specifically claims that the proportion of
improvement in “after-test results” (the population parameter) is 50%, as expected
under normal conditions.
8.1.5 Outcomes of Hypothesis Test
A hypothesis test begins with the assumption that the null hypothesis is true. We
then examine the data to see if there is sufficient reason to reject this assumption.
Consequently, there are typically only two possible outcomes for any hypothesis test,
summarized as follows. Note that "accepting the null hypothesis" is not an outcome
because the null hypothesis is the initial assumption. The hypothesis test may not
provide sufficient evidence to reject this initial assumption, but it cannot prove the
initial assumption is true.
Null and Alternative hypotheses
The null hypothesis claims a specific value for a population parameter. (It is often
the value expected in the case of no special effect.) It takes the form

null hypothesis: population parameter = claimed value

The alternative hypothesis is the claim that is accepted if the null hypothesis is
rejected.
Two Possible Outcomes of a Hypothesis Test
 Rejecting the null hypothesis, in a case means that we have evidence in
support of the alternative hypothesis.
 Not rejecting the null hypothesis, in a case where we lack sufficient
evidence to support the alternative hypothesis.

8.3 Null Hypothesis: To Reject or Not to Reject

Let us consider an example. A teacher claims that a new teaching method helps 75%
of students pass a test. You study a sample of 100 students and find that 85 of them
passed the test. Now, you need to decide if this is enough evidence to reject the null
hypothesis.

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 According to the null hypothesis, the true proportion of students passing the test with
the new method is 75% (25% is not counted in it because % means 100). Here, 75%
comes from the teacher’s claim. The results showed that by random chance, in a sample
of 100 students, 85 might pass.
Through statistical methods, you can calculate the probability of getting 85 students
passing by chance if the true proportion (remember the teacher’s claim of 75%) is indeed
75%. So, if we see 15 students out of each group of 100 students for chance because 85
out of 100 passed the test, then it is possible that this probability of chance turns out to
be about 2 in 1000 (or 0.002) i.e., 15×10= 150/2=75. So, if the results showed that 85
out of 100 passed the test, then 15×10= 150 students may pass the test by chance.
Keeping the significance level at 0.05 means that for an event to be statistically
significant, the probability of it happening by chance needs to be less than 5 in 100 (or
0.05). Since 0.002 is less than 0.05, the result is statistically significant at the 0.05 level.
Therefore, you have good reason to reject the null hypothesis and support the idea that
the new teaching method helps at least 75% of students pass the test.
Table 8.7 Decisions About Hypothesis Testing Based on Significance Value
We decide the outcome of a hypothesis test by comparing the actual sample result
(mean or proportion) to what we would expect if the null hypothesis is true. We use a
significance level, often denoted by 𝛼 (pronounced as alpha) to determine how
confident we need to be before rejecting the null hypothesis.

 Strong Evidence (0.01 Level): If the chance of observing a sample result at

least as extreme as the observed result is less than 1 in 100 (or 0.01), the test
is significant at the 0.01 level. This means that there is strong evidence for
rejecting the null hypothesis and accepting the alternative hypothesis.
 Moderate Evidence (0.05 Level): If the chance of observing a sample result
at least as extreme as the observed result is less than 1 in 20 (or 0.05), the test
is significant at the 0.05 level. It indicates moderate evidence for rejecting the
null hypothesis.
 Insufficient Evidence: If the chance of observing a sample result at least as
extreme as the observed result is greater than 1 in 20, the test is not significant.
In this case, we do not have sufficient grounds for rejecting the null
hypothesis.

These guidelines help us interpret the results of hypothesis tests and make
informed decisions about whether to accept or reject the null hypothesis based on
the level of evidence provided by the data.

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8.4 Confidence Interval
Hypothesis testing involves making a claim or hypothesis about a population
parameter, like the average height of all students, and then testing whether the
evidence from a sample supports or contradicts that claim.
Let us look at an example. For a hypothesis “the average height of all students
in the university is 175 cm”, you collect a sample of students, measure their heights,
and calculate the sample mean. Now, you want to see if the evidence from your
sample supports or contradicts your hypothesis. This confidence interval is like a
guess at a range of heights that you are confident the true average height falls within.
For example, if you find that the average height of your sample is 170 cm with a
95% confidence interval of ±5 cm, it means you're pretty confident that the true
average height of all students in the university is somewhere between 165 cm and
175 cm. So, in simple terms, a confidence interval is like saying, "I think the answer
is probably somewhere between X and Y, but I'm not 100% sure." It helps you
understand the uncertainty in your estimate.

8.5 Type I and Type II Error

Type I and Type II errors are important concepts that relate to the decisions made
regarding the null hypothesis.
(i) Type I Error (False Positive): Type I error occurs when the null hypothesis
is incorrectly rejected. It means indicating that there is evidence for an effect
or difference when, in reality, there is none. For example, in a clinical trial,
concluding that a new drug is effective in treating a disease when it actually
has no effect on the disease (i.e., the null hypothesis of no difference between
the drug and placebo is rejected when it is true).
Type I errors are associated with alpha (𝛼), the significance level chosen for
the hypothesis test. The probability of making a Type I error is equal to the
chosen significance level (𝛼). For example, if the significance level is set at
0.05, there is a 5% chance of committing a Type I error.

(ii) Type II Error (False Negative): Type II error occurs when the null hypothesis
is incorrectly not rejected It means indicating that there is no evidence for an
effect or difference when, in reality, there is one. For example, in a clinical
trial, failing to conclude that a new drug is effective in treating a disease when
it actually is treating the drug (i.e., the null hypothesis of no difference between
the drug and placebo is not rejected when it is false). Type II errors are
influenced by factors such as sample size, effect size, and variability in the
data.

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 In summary, Type I and Type II errors represent the risks associated with
hypothesis testing. While Type I errors involve incorrectly rejecting the null
hypothesis, Type II errors involve incorrectly failing to reject the null hypothesis.
Researchers must consider these errors when interpreting the results of hypothesis tests
and strive to minimize both types of errors by carefully selecting significance levels
and designing studies with adequate sample sizes.

Summary
The unit on hypothesis testing has provided a comprehensive understanding of
this fundamental statistical concept and its significance in research and decision-
making processes. A hypothesis serves as a tentative explanation or prediction about
the relationship between variables or the characteristics of a population. There are two
forms of hypotheses: the null hypothesis, which represents the default assumption or
no difference/effect, and the alternative hypothesis, which suggests a difference or
effect of a phenomenon.
Throughout the unit, we delved into the significance of hypotheses in scientific
inquiry, emphasizing their role in guiding research questions, designing experiments,
and drawing conclusions based on empirical evidence. Hypotheses serve as the
foundation for hypothesis testing, a systematic procedure used to assess the validity of
hypotheses using sample data.
The procedure of hypothesis testing involves comparing the actual value with an
appropriate significance level (alpha) (check table 8.6). Making decisions about
whether to reject or fail to reject the null hypothesis relies on comparing the
significance value to the critical value or predetermined significance level. If it is less
than the significance level, we reject the null hypothesis. Conversely, if the significance
value exceeds the significance level, we fail to reject the null hypothesis, suggesting
that there is insufficient evidence to support the alternative hypothesis.
In summary, hypothesis testing is a vital tool in scientific research, enabling
researchers to draw meaningful conclusions, test theoretical predictions, and make
evidence-based decisions. By mastering the concepts and procedures of hypothesis
testing, researchers can enhance the rigor and validity of their studies, contributing to
the advancement of knowledge in their respective fields.

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Multiple-Choice Questions
1) What is the primary purpose of hypothesis testing?
a) To collect data for a study
b) To determine the significance of a claim
c) To analyze raw data
d) To eliminate Type I errors
Answer: b) To determine the significance of a claim

2) Which of the following best defines a null hypothesis?

a) A statement predicting an effect or relationship
b) A statement proving the researcher's bias
c) A statement focusing on variables' characteristics
d) A statement asserting no effect or relationship exists
Answer: d) A statement asserting no effect or relationship exists

3) What is the confidence interval used for in hypothesis testing?

a) To estimate the range of possible sample sizes
b) To calculate the likelihood of a Type I error
c) To define the range where the true population parameter is likely to fall
d) To assess the variability in test statistics across samples
Answer: c) To define the range where the true population parameter is likely to
fall
4) A Type I error occurs when:
a) A true null hypothesis is rejected
b) A false null hypothesis is accepted
c) The sample size is too small
d) The confidence interval is incorrectly calculated
Answer: a) A true null hypothesis is rejected

5) What does a p-value represent in hypothesis testing?

a) The probability of the null hypothesis being true
b) The confidence interval's midpoint
c) The probability of making a Type II error
d) The probability of observing data as extreme as the sample data, assuming the
null hypothesis is true
Answer: d) The probability of observing data as extreme as the sample data,
assuming the null hypothesis is true

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6) Which of the following is an alternative hypothesis?
a) The population mean differs from the hypothesized value
b) The population mean is equal to the hypothesized value
c) There is no difference between the sample and population mean
d) The observed data aligns with the null hypothesis
Answer: a) The population mean differs from the hypothesized value

7) A Type II error occurs when:

a) A true null hypothesis is rejected
b) The sample size is too large
c) A false null hypothesis is accepted
d) The significance level is too high
Answer: c) A false null hypothesis is accepted

8) What does the term "statistical significance" mean?

a) The data is perfectly measured
b) The results are unlikely due to chance
c) The null hypothesis is always true
d) The sample size is sufficient
Answer: b) The results are unlikely due to chance

9) Which step comes first in hypothesis testing?

a) Collecting data
b) Stating the null and alternative hypotheses
c) Setting the significance level
d) Performing a statistical test
Answer: b) Stating the null and alternative hypotheses

10) What is the typical significance level used in hypothesis testing (in social sciences)?
a) 5%
b) 10%
c) 20%
d) 1%
Answer: a) 5%

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Short-Answer Questions
1) What is the difference between a null hypothesis and an alternative hypothesis?
2) Define a confidence interval in your own words.
3) Explain the consequences of a Type I error in hypothesis testing.
4) What role does a p-value play in deciding to accept or reject a hypothesis?
5) Why is it important to understand the outcomes of a hypothesis test?
Subjective Questions
1) Describe the steps involved in hypothesis testing. Provide an example for each
step.
2) Discuss the significance of confidence intervals in hypothesis testing and how
they impact decision-making.
3) Compare and contrast Type I and Type II errors, and explain their implications in
real-world research.
4) Explain the importance of setting an appropriate significance level in hypothesis
testing.
5) Discuss how hypothesis testing can be applied in educational research. Provide
relevant examples.
6) In a study of children injured in automobile crashes (American Journal of Public
Health, Vol. 82, No. 3), those wearing seat belts had a mean stay of 0.83 days in an
intensive care unit. Those not wearing seat belts had a mean stay of 1.39 days. The
probability of this difference in means occurring by chance turns out to be less than
1 in 10,000. Would it be reasonable to conclude that seat belts reduce the severity
of injuries? Explain.
7) In a study comparing the effectiveness of two pain relief medications, participants
who received Medication A reported an average pain reduction of 40%, while those
who received Medication B reported an average reduction of 30%. The probability
of this difference occurring by chance is less than 1 in 1000. Can we conclude that
Medication A is more effective in reducing pain than Medication B?
8) A company introduces a new energy-efficient light bulb and claims that it lasts an
average of 5000 hours longer than the standard light bulb. After testing 1000 of
these new light bulbs, it is found that they indeed last an average of 5000 hours
longer with a probability of this difference occurring by chance being less than 1 in
10000. Can we conclude that the new light bulb lasts longer than the standard one?
9) A school district implements a new teaching method and claims that it increases
student test scores by at least 15%. After implementing the method, they find that
the average test scores increased by 20% with a probability of this difference
occurring by chance being less than 1 in 5000. Is this result significant at the 0.05
level? Can we conclude that the new teaching method improves test scores?
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10) A study investigates the impact of a new exercise program on weight loss.
Participants who followed the new program lost an average of 10 pounds, while
those who did not lost an average of 5 pounds. The probability of this difference
occurring by chance is less than 1 in 100. Is this result significant at the 0.05 level?
Can we conclude that the new exercise program leads to greater weight loss?
11) An online retailer introduces a new website design and claims that it increases
average customer spending by at least 20%. After the redesign, they found that
average spending increased by 25% with a probability of this difference occurring
by chance being less than 1 in 500. Is this result significant at the 0.05 level? Can
we conclude that the new website design increases customer spending?
12) A hospital introduces a new surgical technique and claims that it reduces recovery
time by at least 30%. Patients who underwent the new technique had an average
recovery time of 5 days, while those who underwent the standard technique had an
average recovery time of 7 days. The probability of this difference occurring by
chance is less than 1 in 1000. Is this result significant at the 0.01 level (in medical
sciences, we use 0.01 because we need to be 99% sure about the study results as
error-free)? Can we conclude that the new surgical technique reduces recovery
time?
13) A study compares the effectiveness of two study techniques on exam performance.
Students who used Technique A had an average score of 85%, while those who
used Technique B had an average score of 80%. The probability of this difference
occurring by chance is less than 1 in 100. Can we conclude that Technique A is
more effective than Technique B?
14) An experiment investigates the effect of a new drug on blood pressure. Patients
who took the new drug experienced an average decrease in blood pressure of 10
mmHg, while those who took the standard drug experienced a decrease of 5 mmHg.
The probability of this difference occurring by chance is less than 1 in 1000. Can
we conclude that the new drug is more effective in reducing blood pressure?
15) A study examines the impact of a new marketing strategy on sales revenue. After
implementing the strategy, the company experienced a 15% increase in revenue
compared to the previous year, with the probability of this difference occurring by
chance being less than 1 in 5000. Can we conclude that the new marketing strategy
increases sales revenue?
16) A research study compares the effectiveness of two diet plans on weight loss.
Participants who followed Diet Plan A lost an average of 8 pounds, while those
who followed Diet Plan B lost an average of 6 pounds. The probability of this
difference occurring by chance is less than 1 in 100. Is this result significant at the

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 0.05 level? Can we conclude that Diet Plan A is more effective in promoting weight
loss?
Activities
1) Write down the null and alternative hypotheses for the following scenarios:
a. A researcher wants to test if exercising daily reduces stress levels compared
to not exercising.
b. A company claims that its new energy drink increases physical performance.
c. A teacher believes that using digital tools improves students’ test scores.
2) For each of the following situations, identify and describe what a Type I and Type
II error would look like:
Situation Null hypothesis Type I Error Type II Error
A smoke detector is
installed to alert for
fires.
A weather prediction
system forecasts rain
A court trial determines
whether an accused
person is guilty.

3) Consider an example. A restaurant manager estimates that the average waiting

time for customers is 15 minutes, with a confidence interval of ±5 minutes. This
means the waiting time is likely between 10 and 20 minutes. Keeping in view this
example, for each situation below, describe the confidence interval and explain
what it means:
a) A survey estimates that 60% of customers prefer online shopping, with a
confidence interval of ±10%.
b) A thermometer shows a body temperature of 98.6°F, with a confidence
interval of ±1°F.
c) A company predicts that its monthly revenue will be $50,000, with a
confidence interval of ±$5,000.
4) Consider an example. Scenario: A school tests whether a new teaching method
improves students’ grades. The hypothesis testing outcomes are as follows:
 True Positive: The method actually improves grades, and the test detects it.
 False Positive: The method does not improve grades, but the test says it does.
 True Negative: The method does not improve grades, and the test correctly
says so.

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  False Negative: The method actually improves grades, but the test fails to
detect it.
Keeping in view this example, for each of the following cases, write the outcome
as True Positive, False Positive, True Negative, or False Negative:
a) A drug effectively treats a disease, and the clinical trial confirms it.
b) A new app claims to reduce screen time but actually does not. However, the
study suggests it does.
c) A noise-reduction headphone works as intended, and testing validates its
effectiveness.
d) A study suggests a diet helps in weight loss, but it actually has no effect.

REFERENCES

Briggs, B.(2015). Using and understanding Mathematics: A quantitative reasoning

approach. USA: Pearson Education Inc.

Cottrell, S. (2015). Critical Thinking Skills. New York: Palgrave Macmillan Ltd.

Starkey, L. (2004). Critical thinking skills success in 20 minutes a day. New York:
Learning Express, LLC.

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Unit–9

PROBABILITY

Written by: Dr. Mubeshera Tufail

Reviewed by: Dr. Shahbaz Hamid

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 INTRODUCTION

Probability is the study of chance and uncertainty. It helps us understand and predict
random events in everyday life. This unit introduces the basic ideas of probability, such as
events and outcomes. You will learn how to calculate the chances of different events
happening.
We will also explore how probabilities can be combined to solve more complex
problems. The law of large numbers will show you how results become more predictable
when repeated many times. Probability also helps in assessing risks, which is useful for
making better decisions. The unit covers counting methods like permutations and
combinations. These tools help solve tricky probability problems. You will also learn how
probability explains coincidences and surprising events.
This unit connects to quantitative reasoning by teaching you how to analyze data, think
logically, and make decisions based on numbers.

Learning Outcomes
After the successful completion of the unit, the students would be able to:
1. Understand the fundamental concepts of probability including events, outcomes,
and probabilities.
2. Calculate probabilities for simple and compound events, and combine probabilities
using appropriate techniques.
3. Explain the law of large numbers and its implications for understanding
probability.
4. Assess risks effectively using probability for making informed decisions based on
probability assessments.
5. Apply advanced counting techniques like permutations and combinations to solve
probability problems.

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9.1 Fundamentals of Probability
9.1.1 Event and its Outcomes
Outcomes are the most basic possible results of observations or experiments. For
example, if you toss two coins, one possible outcome is HT (one coin’s head and the
other one tail) and another possible outcome is HH (heads of both coins). Other
possible outcomes of this coin tossing are given below. An event consists of one or
more outcomes that share a property of interest. For example, if you toss two coins
and count the number of heads, the outcomes HT and TH both represent the same
event of 1 head and 1 tail.
Note: Please note that H stands for head and T for tail.
Consider a scenario where two coins are tossed. There are four potential ways the
coins can land: both tails, tails followed by heads, heads followed by tails, or both
heads. These outcomes are represented as TT, TH, HT, and HH respectively. It is
important to note that the order of the outcomes matters; for example, TH is distinct
from HT. Now, if we focus solely on the number of heads, we can group together
outcomes that have the same number of heads. For instance, TH and HT both have
one head, so they represent the same event. This concept of an event encompasses one
or more outcomes that share a common characteristic of interest—in this case, the
number of heads. So, by looking at the possible outcomes, we observe that there are
only three distinct events when counting the number of heads on two coins: no heads
(0H), one head (1H), and two heads (2H).

Figure 27 Four Possible Outoces of Two Coins Tossing at the Same Time

Example
Let us consider another example. Suppose there are two students in a class. They
took a final examination. There are two possible results: pass (P) and fail (F). So,
there are three possible outcomes for the event “result of the final examination”.
1. Both students pass. (PP)
2. One student fails and the other student pass. (there are two outcomes FP
and PF because the first student may fail and the second student may pass,
and vice versa.)
3. Both students fail.

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 There is another way to explain this case. The event with “no pass (no P)” is only
one i.e., when both students fail. The event with “no fail (no F)” is one when both
students pass the examination. The event with one pass has two outcomes: when the
first student fails and the second student passes the examination, and when the first
student passes and the second student fails in the examination.

Table 9.8 List of probabilities for tossing a coin

Student 1 Student 2 Outcome
Pass (P) Pass (P) PP
Fail (F) Pass (P) FP
Pass (P) Fail (F) PF
Fail (F) Fail (F) FF

9.1.2 Finding Probabilities

The probability of an event, expressed as P (event), is always between 0 and 1
(inclusive). A probability of 0 means the event is impossible and a probability of 1
means the event is certain.
Consider the outcomes of tossing a coin. There are two possible outcomes of
tossing a coin one time: head (H) and tail (T). Let us assume that the tossing of a coin
is a fair event where the likelihood of it landing heads (H) or tails (T) is equal. In
colloquial terms, we often say that the chance of the coin landing heads in a single toss
is "50-50," indicating that heads and tails are expected with equal probability, 50%
each. However, when dealing with calculations, it is more precise to express this
probability as a fraction.
With one toss of the coin, there are two equally probable outcomes: either head
(H) or tails (T). If our event of interest is the coin landing heads (H), then the
probability of this event is 1 out of 2, written as 1/2, because heads is one of the two
equally possible outcomes (H and T). Using the notation P(event) to represent
probability, we express the probability of heads as follows:

𝑃(𝐻) = 1/2 = 0.5

It means that "the probability of heads equals one-half, or 0.5." Similarly, we can
express the probability of the coin landing tails as:

𝑃 (𝑇) = 1/2 = 0.5

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 Notice that since only heads (H) and tails (T) are possible outcomes, it is certain
that we will get either heads or tails when we toss the coin. Thus, the probability of
the event "coin lands either heads or tails" is 2 out of 2, or 1:

𝑃 (coin lands either heads or tails) = 2/2 = 1

In other words, an event that is certain to occur has a probability of 1. Conversely,

any event that does not include heads (H) or tails (T) as outcomes has a probability of
zero. For example, the probability of the event "coin remains balanced on its edge
forever" is zero.
In general, probabilities always range between 0 and 1. A probability of 0
indicates that an event is impossible, while a probability of 1 is certainty of the event.
Values in between represent different degrees of likelihood. The scale of probability
values, as shown in Figure 7.2, illustrates various degrees of certainty, from
impossible to certain.

Figure 9.28 Scale showing various degrees of certainty as expressed by probabilities.

Now that we understand how to express probabilities, we can consider how to calculate
or estimate them. There are three basic techniques for finding probabilities, which we call
the theoretical method, the relative frequency method, and the subjective method.
Techniques for Finding Probability
1) Theoretical Probability: Theoretical probability is based on the assumption
that all outcomes are equally likely. It is calculated by dividing the number of
ways an event can occur by the total number of possible outcomes.
Example: For a single toss of a fair coin, there are two possible outcomes: heads
(H) or tails (T). The theoretical probability of getting heads is:

𝑃 (𝐻) = 1/2 = 0.5

Similarly, the probability of getting tails is also:

𝑃 (𝑇)= 1/2 = 0.5

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 2) Relative Frequency Probability: Relative frequency probability is based on
actual observations or experiments. It is the ratio of the number of times an
event occurs to the total number of trials.
Example: Suppose you toss a fair coin 100 times and record the outcomes. If the
coin lands on heads 48 times and on tails 52 times, the relative frequency
probability of getting heads is:
𝑃(𝐻)=48/100 =0.48

The relative frequency probability of getting tails is:

𝑃(𝑇)=52/100 =0.52

3) Subjective Probability: Subjective probability is an estimate based on

personal judgment, experience, or intuition rather than on formal calculations or
experiments.
Example: Imagine you are about to flip a coin that you suspect might be biased
based on your past experience (perhaps you noticed it seems to land on heads
more often). Based on your intuition, you might estimate that the probability of
getting heads is higher than 50%. In case, you toss the coin 100 times, you might
say that for the head of the coin:
𝑃(𝐻)= 60/100 = 0.6

Thus, the subjective probability of getting tails would be:

𝑃(𝑇)= 40/100 = 0.4

This estimate is not based on precise calculations or a large number of trials

but rather on your personal judgment. These examples show how the concept of
probability can be applied to the simple act of tossing a coin, illustrating the
differences between theoretical probability (equal likelihood), relative frequency
probability (observed outcomes), and subjective probability (personal judgment).

Making a Probability Distribution

A probability distribution represents the probabilities of all possible events of interest. To
make a table of a probability distribution (see table 9.8 for example):
Step 1. List all possible outcomes. Use a table or figure if it is helpful.
Step 2. Identify outcomes that represent the same event. Find the probability of each event.
Step 3. Make a table in which one column lists each event and another column lists each
probability. The sum of all the probabilities must be 1.

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9.2 Combining Probabilities
9.2.1 And Probability for Independent Events
Two events are independent if the occurrence of one event does not affect the
probability of the other event. Consider two independent events A and B with individual
probabilities P(A) and P(B). The and probability that A and B both occur is
P (A and B) = P(A) × P(B)
Consider the example of the tossing of two coins here. This principle can be
extended to any number of independent events. For example, the and probability of
three independent events (e.g., tossing of three coins) A, B, and C is
P (A and B and C) = P(A) × P(B) × P(C)
For example, if we toss three coins at the same time, there is a possibility of getting a
head for all three coins. How many times we can receive this outcome? We know that
the probability of heads on each individual coin is 1/2. Therefore, we can find the “and
probability” of 3 heads by multiplying the three individual probabilities together. Let
us see it.

P (3 head) = P (head on coin 1) × P (head on coin 2) × P(head on coin 3)

= 1/2 × 1/2 × 1/2 = 1/8

There is a probability of only one event out of 8 where we can get the head of all three
coins.

9.2.2 And Probability for Dependent Events

Two events are dependent if the outcome of one event affects the probability of
the other event. The “and probability” that dependent events A and B both occur is:

P(A and B) = P(A) × P(B given A)

where P(B given A) means “the probability of event B given the occurrence of event
A.” This principle can be extended to any number of dependent events. For example,
the “and probability” of three dependent events A, B, and C is:

P(A and B and C) = P(A) × P(B given A) × P(C given A and B)

The key steps are to first find the probability of the first event, then the conditional
probability of the second event given the first, and finally multiply these probabilities
to find the combined probability. In this case, drawing two red marbles in a row from
a bag of 8 marbles (5 red and 3 blue) without replacement has a probability of 5/14.

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Example: You have a bag containing 5 red marbles and 3 blue marbles. You draw
two marbles sequentially without replacement, and you want to find the probability
that both marbles drawn are red.

Step-by-Step Calculation
First Event (A): Drawing a red marble on the first draw.
Second Event (B): Drawing a red marble on the second draw given that the first
marble drawn was red.

Step 1: Calculate the Probability of the First Event (A)

 Total marbles in the bag: 5 red + 3 blue = 8 marbles.
 Number of red marbles: 5.
 Probability of drawing a red marble first (Event A)
𝑃(𝐴)=5/8

Step 2: Calculate the Probability of the Second Event (B) Given the First Event (A)
 After drawing one red marble, there are now 4 red marbles left and a total
of 7 marbles remaining in the bag.
 Probability of drawing a red marble second given that the first marble was
red (Event B given A):
𝑃(𝐵∣𝐴)=4/7

Step 3: Calculate the Combined Probability of Both Events (A and B)

 The probability that both events (drawing two red marbles in a row)
occur is found by multiplying the probability of the first event by the
conditional probability of the second event:
𝑃(𝐴 and 𝐵) =𝑃(𝐴)×𝑃(𝐵∣𝐴)

 Substitute the probabilities calculated in Steps 1 and 2:

𝑃(𝐴 and 𝐵)=5/8 × 4/7

 Perform the multiplication:

𝑃(𝐴 and 𝐵)=(5×4)/(8×7) = 20/56 = 5/14

Hence, the probability that both marbles drawn are red is:
𝑃(both red)=5/14

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 9.2.3 Either/Or Probability for Non-overlapping Events
Two events are non-overlapping if they cannot occur together. If A and B are non
overlapping events, the probability that either A or B occurs is
P(A or B) = P(A) + P(B)
This principle can be extended to any number of non-overlapping events. For example,
the probability that either event A, event B, or event C occurs is
P1A or B or C2 = P1A2 + P1B2 + P1C2
Non-overlapping events are those that cannot occur simultaneously. For instance,
when tossing a coin, the outcomes "heads" and "tails" cannot occur together because
the coin cannot land on both sides at once. In a Venn diagram, these events are
represented by non-overlapping circles. The probability of either mutually exclusive
event occurring is the sum of their individual probabilities. For example, with a fair
coin, the probability of landing heads is 1/2 and tails is ½. Thus, the probability of
getting either heads or tails in a single toss is 1/2 + 1/2 = 1 indicating that one of the
two outcomes will always happen.

9.3 Law of Large Numbers

The law of large numbers applies to a process for which the probability of an event
A is P(A) and the results of repeated trials do not depend on the results of earlier trials
(they are independent). It states: If the process is repeated through many trials, the
proportion of the trials in which event A occurs will be close to the theoretical
probability P(A). The larger the number of trials, the closer the proportion should be to
P(A). (Remember theoretical probability and relative frequency probability)
For example, we want to predict the probability of one side containing 1 point out
of six sides of a side i.e., 1, 2, 3, 4, 5, 6, as shown in figure 9.29. To demonstrate, we
can use a computer to simulate rolling a dice 5000 times. Based on the results of this
simulation, we plot a graph between a number of rolls (x-axis) and the proportion of
rolls in a 1 (y-axis), as shown in the figure given below. Initially, the proportion of 1s
varies significantly because the number of rolls is small, and random fluctuations have
a larger impact. As the number of rolls increases, the proportion of 1s stabilizes and
approaches the theoretical probability of 0.167.
This behavior aligns with the law of large numbers, which predicts that the
experimental probability will converge to the theoretical probability as the number of
trials grows. The simulation graphically shows that the proportion of 1s becomes more
stable and close to 0.167 with a larger number of rolls.

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 Figure 9.29 Graph between the number of rolls (x-axis) and proportion of rolls in a
1 (y-axis)

9.4 Assessing the Risk

Assessing risk involves evaluating the potential benefits and drawbacks of a
decision to determine whether the advantages outweigh the possible negative
outcomes. This process requires understanding the probabilities and impacts of
different scenarios. For example, when deciding whether to use a prescription
medication, one must consider the likelihood and severity of side effects against the
medication's effectiveness in treating a condition. Similarly, choosing to eat processed
foods involves weighing the convenience and taste against the long-term health risks.
By quantifying these risks and benefits, individuals can make informed decisions that
align with their values and circumstances, balancing immediate gains with potential
future consequences. We make tradeoffs between the benefits and risks of a decision.
By understanding probability, we can quantify risk and make more informed decisions
about these tradeoffs.

Example 1: Using Prescription Medications

Prescription medications are a common part of modern healthcare, offering
significant benefits such as treating illnesses, managing chronic conditions, and
improving overall quality of life. However, they also come with potential risks and
side effects.

Tradeoff: The decision to use prescription medications involves balancing the

benefit of effectively managing a health condition (like hypertension) against the risk
of potential side effects (such as dizziness, fatigue, or severe reactions).

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 Benefit: A medication can effectively manage a chronic condition like hypertension,
reducing the risk of heart attack and stroke.

Risk: The same medication might have side effects like dizziness, fatigue, or even
more severe reactions in rare cases.

Most people, after consulting with their doctor, decide that the benefits of
controlling a serious condition outweigh the potential side effects, making the
medication worth taking.
Example 2: Eating Processed Foods
Processed foods are convenient, often tasty, and have a longer shelf life compared
to fresh foods. They are a staple in many people's diets because of their accessibility
and time-saving qualities.

Tradeoff: The choice to consume processed foods involves weighing the benefit of
convenience and time-saving against the risk of potential health issues (such as
obesity, diabetes, and heart disease).

Benefit: Convenience and time-saving, making it easier to prepare meals quickly,

which is crucial for people with busy schedules.

Risk: Regular consumption of highly processed foods can lead to health issues such
as obesity, diabetes, and heart disease due to high levels of sugar, salt, and unhealthy
fats.
People often balance these tradeoffs based on their lifestyle, choosing processed
foods for convenience despite knowing the long-term health risks, while trying to
incorporate healthier options when possible. They can choose fresh food over
processed food keeping in view the risk and benefits involved in this decision.

So, to assess the risk and probability of a scenario, you can follow some steps:
 Gather relevant data or use provided probabilities.
 Identify the factors influencing the decision (e.g., historical weather patterns,
average time for checkout).
 Calculate the probability of the event occurring.
 Discuss how this probability affects your decision-making process and the
potential tradeoffs involved.

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 Activities:

Example 1: Weather Forecast: Deciding whether to carry an umbrella.

Question: What is the probability that it will rain tomorrow?
Activity: You can use weather forecasts which often provide probabilities, like
"there is a 70% chance of rain tomorrow." You can discuss how this probability
affects your decision to carry an umbrella and the potential consequences of getting
wet if it rains versus carrying extra weight if it does not.

Example 2: Choosing a Line at the Grocery Store: Choosing the fastest checkout
line.
Question: What is the probability that the line with fewer people will be faster?
Activity: Students can observe and collect data on the number of items in each
person's cart in different lines. They can estimate the time it takes for a single item
to be scanned and calculate the total expected time for each line. They will then use
this data to assess the probability that the shorter line will be faster.

Example 4: Planning a Study Schedule: Preparing for a surprise quiz.

Question: What is the probability that there will be a surprise quiz in the next class?
Activity: Students can review the teacher’s past behavior regarding surprise quizzes
(e.g., if the teacher has given a surprise quiz 2 out of the last 10 classes). They can
use this historical data to estimate the probability of a surprise quiz in the next class.
They will then discuss how this probability influences their decision to prepare for a
quiz or focus on other assignments.

9.5 Counting and Probability

Understanding theoretical probabilities often depends on being able to count different
possible outcomes correctly. For simple problems, this might just mean counting things
directly or using basic multiplication to find the total number of possible results. However,
for more complicated problems, we need more advanced methods to count correctly. These
advanced methods help us solve a wider variety of probability questions and help us
understand why seemingly surprising events (like coincidences) actually happen more
often than we might think. For this purpose, besides the multiplication principle (topic 9.2
of this unit), there are three techniques given below to count the probability of certain
outcomes.
1) Understanding Arrangements with Repetition
2) Permutations
3) Combinations

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 9.5.1 Understanding Arrangements with Repetition
Arrangements with repetition refer to the different ways in which items can
be arranged when repetition of items is allowed, allowing the same item to be
used multiple times. This concept is often explored in the context of
probability. If we have 𝑛 items and we want to make arrangements of 𝑟 items
where repetition is allowed, the number of possible arrangements is given by:

𝑛𝑟

Examples
Example 1: Create a 4-character password using digits (0-9). How many possible
passwords can be created?
Solution: Each of the 4 characters can be any of the 10 digits. Therefore, the number of
possible arrangements is:
104 = 10,000
So, there are 10,000 possible 4-digit passwords.

Example 2: Choosing a 3-scoop ice cream cone from 5 available flavors where
repetition is allowed (you can choose the same flavor more than once). How many
different combinations of 3 scoops can you have?
Solution: Each of the 3 scoops can be one of the 5 flavors. Therefore, the number of
possible arrangements is:
53 = 125
So, there are 125 different possible combinations of 3-scoop ice cream cones.

Example 3: Generating a 6-digit PIN code using digits (0-9). How many possible PIN
codes can be created?
Solution
Each of the 6 digits can be any of the 10 digits. Therefore, the number of possible
arrangements is:

106 = 1,000,000

9.5.2 Permutations
Permutations occur when we are selecting items from a single group, where
each item can only be chosen once, and the order of selection matters. For
example, if we're arranging letters in a word, like ABCD, it's different from
arranging them as DCBA.

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 If we are using every item in the group for each arrangement, then the total
number of possible arrangements, or permutations, with
𝑛 items are given by 𝑛!, where 𝑛! = 𝑛 × (𝑛 − 1) × (𝑛 − 2) ×…× 2 ×1. This is
read as "n factorial."
Examples
Example 1: Creating a Password:
For a 4-digit password using numbers 0-9, there are 10 options for the first digit, 9
options for the second digit (since we cannot repeat the first digit), 8 options for the
third digit, and 7 options for the fourth digit. Using the formula, 𝑛! we calculate
10×9×8×7=5040. So, there are 5040 possible passwords.

Example 2: Arranging Books on a Shelf

Here, we have 5 different books to arrange. Using the formula, 𝑛!, we calculate
5!=5×4×3×2×1=120. So, there are 120 different ways to arrange the books on the shelf.

Example 3: Planning a Dinner Menu:

In this scenario, we have 6 options for the appetizer, 5 options for the main course, and
4 options for dessert. Using the formula, 𝑛!, we calculate 6×5×4=120. So, there are 120
possible combinations for the dinner menu.

9.5.3 Combinations
Combinations occur whenever all selections come from a single group of
items, no item may be selected more than once, and the order of arrangement
does not matter (for example, ABCD is considered the same as DCBA).

Example: Choosing Fruits

You have 5 different fruits and you want to choose 3 of them for a fruit salad.
The order in which you choose the fruits does not matter (apple, banana, cherry
is the same as cherry, banana, apple).
Solution: Using the formula for combinations:
𝐶(5,3) = 5! = 5×4×3×2×1 = 120 =10
3!(5−3)! (3×2×1)(2×1) 6×2
So, there are 10 different ways to choose 3 fruits from 5.

9.6 Probability and Coincidence

The word coincidence literally refers to one or more incidents happening together,
such as events happening in the same place or at the same time. Although a particular
coincidence may be highly unlikely, some similar coincidence may be extremely likely
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 or even certain to occur. In general, this means that coincidences should be expected to
occur, with their likelihoods dictated by the laws of probability.
There are various types of coincidences. For example, it might be surprising to find
that two guests at a dinner party share the same birthday, or that a friend's mother met
your mother during a trip to China. You could flip a coin and get heads ten times in a
row, or have a dream that appears to predict a future event. It is natural to attribute these
instances to something beyond mere chance. However, the principles of probability
explain that such coincidences are expected to occur, even though their specific
manifestations are unpredictable.
For example, most people think that the probability of finding another person with
their exact birthday in a group of 25 people is low. This is a specific coincidence.
However, the probability of finding any two people in the group who share a birthday,
which is a more general coincidence, is quite high—almost 3 in 5 (or 57%).
Summary
In this unit, we explored the basics of probability, covering key concepts like events,
outcomes, and how to calculate and combine probabilities. We learned about the law of
large numbers, which shows that the results of many trials get closer to the expected
probability. We also looked at assessing risk, using probability to make informed decisions
by weighing potential risks and benefits. The unit included counting techniques like
permutations and combinations to solve problems that are more complex. Finally, we
discussed coincidences showing that many seemingly surprising events are actually
expected according to the laws of probability. Overall, this unit provided a solid foundation
for understanding how probability helps us make sense of randomness and uncertainty in
daily life.
Multiple-Choice Questions
1) What is the probability of an event that is certain to happen?
a) 0
b) 0.5
c) 1
d) 0.3
Answer: c) 1
2) What is the term for the set of all possible outcomes of an event?
a) Probability distribution
b) Sample space
c) Event horizon
d) Outcome set
Answer: b) Sample space
3) If a coin is flipped twice, what is the probability of getting two heads?
a) 0.25
b) 0.5
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 c) 0.75
d) 1
Answer: a) 0.25
4) Which law states that the relative frequency of an event approaches its
probability as the number of trials increases?
a) Law of Small Numbers
b) Law of Large Numbers
c) Law of Coincidence
d) Law of Counting
Answer: b) Law of Large Numbers
5) When two events cannot happen at the same time, they are called:
a) Independent events
b) Mutually exclusive events
c) Complementary events
d) Conditional events
Answer: b) Mutually exclusive events
6) What is the probability of rolling a 5 or a 6 on a single die?
a) 1/6
b) 1/3
c) 1/2
d) 2/3
Answer: b) 1/3
7) The probability of an event not happening is called:
a) Complement probability
b) Conditional probability
c) Joint probability
d) Absolute probability
Answer: a) Complement probability
8) If you roll a fair six-sided dice, what is the probability of rolling a 3?
a) 1/6
b) 1/4
c) 1/3
d) 1/2
Answer: c) 1/13
9) Which of the following best describes risk in probability?
a) The likelihood of an event occurring
b) The complement of an event
c) The average of all possible outcomes
d) The chance of a negative outcome
Answer: d) The chance of a negative outcome
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10) If events A and B are independent, then the probability of both events occurring
is given by:
a) P(A) + P(B)
b) P(A | B)
c) P(A ∪ B)
d) P(A) × P(B)
Answer: d) P(A) × P(B)

Short-Answer Questions
1) Define probability in your own words and provide a real-life example.
2) What is a sample space? List the sample space for tossing a coin three times.
3) Explain the difference between independent and mutually exclusive events.
4) What does the Law of Large Numbers imply in probability?
5) How do you calculate the probability of an event not happening? Provide an
example.
Subjective Questions
1) Discuss the significance of understanding probability in everyday decision-
making. Provide examples.

2) Explain the concept of the Law of Large Numbers and its importance in real-world
applications, such as insurance or gambling.

3) Describe the process of combining probabilities. How does it differ for

independent and mutually exclusive events?

4) Analyze the role of probability in assessing risks. Use examples such as weather
forecasting or investment planning.

5) Write an essay explaining the relationship between probability and coincidence,

including examples from daily life.

6) You are planning a trip to a theme park with your friends. You know that there are
5 roller coasters, 8 water rides, and 3 shows to choose from. You want to make
sure you experience at least one roller coaster and one water ride. How many
different combinations of rides can you choose for your day at the theme park?
Using the principles of combinations, how many different ways can you select at
least one roller coaster and one water ride for your day at the theme park?

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7) You are flipping a fair coin three times. What is the probability of getting at least
two heads? Calculate the probability of getting at least two heads when flipping a
fair coin three times, and explain your reasoning.

8) You are a teacher planning a class field trip to a local museum. The museum offers
two different guided tours: one focusing on art history and the other on natural
history. There are 15 students in your class, and you can only book one tour for
the entire group. If 9 students prefer the art history tour and the rest prefer the
natural history tour, which tour would you choose to maximize student
satisfaction? Using the principles of probability, determine the probability of
selecting the tour that maximizes student satisfaction, and explain your decision-
making process.

9) You are considering investing in a stock. You know that historically, there is a
60% chance that the stock will increase in value over the next year. However, you
are also aware that past performance does not guarantee future results. How would
you assess the risk of investing in this stock? Explain how you would use
probability to assess the risk of investing in the stock, considering both the
historical performance and the uncertainty of future outcomes.

10) You are planning a dinner party and want to create a menu with appetizers, main
courses, and desserts. You have 5 appetizer options, 7 main course options, and 4
dessert options. How many different dinner menus can you create? Using the
principles of permutations, determine the total number of different dinner menus
you can create, and explain your reasoning.

11) You and your friends are discussing coincidences. One of your friends mentions
that they recently met someone with the same birthday as them, and they find it
surprising. How would you explain the probability of this coincidence occurring?
Explain how you would use probability to analyze the likelihood of two people
sharing the same birthday and discuss the factors that contribute to this probability.

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Activities
1) Take a standard six-sided die and roll it 30 times. Record the outcome of each roll
in a table. Calculate the relative frequency of each number (1 to 6) appearing.
Compare your results with the theoretical probability of 1/6 for each number. Write
a short paragraph on how your results demonstrate or differ from the Law of Large
Numbers.

2) Toss a coin 50 times and record the outcomes (heads or tails). Calculate the relative
frequency of heads and tails. Does the relative frequency become closer to the
theoretical probability of 0.5 as the number of tosses increases?

3) A factory produces 1,000 items daily. The probability of a defect in an item is 0.02.
Calculate the expected number of defective items in a day. Reflect on how this
information could help the factory management. Write a paragraph on how
probability helps in quality control.

4) Create a scatterplot for the outcomes of a dice roll and coin toss (30 times for coin
and 30 times for dice roll). Record any patterns and answer: Does the scatterplot
show equal likelihood for each outcome? Write your observations in 3-4 sentences.

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 REFERENCES

Briggs, B.(2015). Using and understanding Mathematics: A quantitative reasoning

approach. USA: Pearson Education Inc.

Cottrell, S. (2015). Critical Thinking Skills. New York: Palgrave Macmillan Ltd.

Starkey, L. (2004). Critical thinking skills success in 20 minutes a day. New York:
Learning Express, LLC.

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