Lesson 16.
Interpretation of the
Population Proportion
Learning Competency
At the end of the lesson, the learners should be able
to draw a conclusion about the population proportion
based on the test-statistic value and the rejection
region (M11/12SP-IVf-2).
Objectives
At the end of this lesson, the learners should be able to
do the following:
● Accurately determine the null and alternative
hypotheses in a hypothesis test for population
proportion.
● Correctly illustrate the rejection region and test
statistic value in a hypothesis test for population
proportion.
Objectives
At the end of this lesson, the learners should be able to
do the following:
● Correctly decide whether to reject the null
hypothesis or not in a hypothesis test for
population proportion.
● Accurately interpret the decision in a hypothesis
test for population proportion.
In conducting research, it is
important to interpret the data
that you gather because this will
help you make decisions. For
instance, researchers can gather
information on the percentage
of people who are vaccinated or
not.
For instance, researchers can
gather information on the
percentage of people who are
vaccinated or not. This data can
be helpful in determining if
people can build up immunity
against the virus and if there is a
need to further encourage more
people to have their vaccines.
In conducting research, it is important to interpret the data that
you gather because this will help you make decisions.
However, the data is only useful if you know what it means.
In this lesson, you will learn how to interpret the test statistic of
a population proportion.
Essential Questions
How does the statement of a claim affect the formulation
of statistical hypotheses?
Is rejecting the null hypothesis equivalent to accepting the
alternative hypothesis? Why do you think so?
Learn about It!
Null Hypothesis
This is a statement which states that there is no difference
between a parameter and a specific value. It is denoted by .
Example:
The proportion of students who have pets is 58%.
Learn about It!
Alternative Hypothesis
This is a statement which states that there is a difference
between a parameter and a specific value. It is the negation of
the null hypothesis.
The alternative hypothesis is denoted by .
Learn about It!
Alternative Hypothesis
Example:
The proportion of students who have pets is not 58%.
Learn about It!
One-Tailed Test
This is a type of hypothesis test that makes use of only one
side or tail of the distribution. It can either be a right-tailed or
left-tailed test.
The one-tailed test is used if the alternative hypothesis
contains a “greater than” or “less than” symbols.
Learn about It!
One-Tailed Test
Example:
A one-tailed test, specifically a right-tailed test, should be
used given the following hypotheses:
• The proportion of students who have pets is at most
58%.
• The proportion of students who have pets is more than
58%.
Learn about It!
Two-Tailed Test
This is a type of hypothesis test that makes use of two
opposite sides or tails of the distribution. It is used if the
alternative hypothesis contains the “not equal to” symbol.
This test is used when no assertion is made on whether the
parameter falls within the positive or negative end of the
distribution.
Learn about It!
Two-Tailed Test
Example:
A two-tailed test should be used given following hypotheses:
The proportion of students who have pets is 58%.
The proportion of students who have pets is not 58%.
Learn about It!
Level of Significance
This is the probability of making an error in rejecting the null
hypothesis when it is actually true.
The commonly used levels of significance are 0.05 and 0.01.
Learn about It!
Level of Significance
Example:
A significance level of 𝛼 = 0.01 means that there is a 1% chance
of rejecting a true null hypothesis.
Learn about It!
Critical Values
There are -scores that are boundaries of the rejection region
Note that in a hypothesis test for population proportion, we
use the -statistic.
Learn about It!
Critical Values
Example:
The table below shows the critical values of and
.
Level of Significance
Type of Test
One-tailed
Two-tailed
Learn about It!
Decision in Hypothesis Testing
When the test statistic value falls in the rejection region, we
reject the null hypothesis. Otherwise, we do not reject the null
hypothesis.
Learn about It!
Decision in Hypothesis Testing
Example:
In a two-tailed test at , the rejection region is the area to the
left of and the area to the right of . If the test statistic value is
less than or greater than , then the null hypothesis is rejected.
Learn about It!
Decision in Hypothesis Testing
Example:
But if the test statistic value is between 𝑧 = −1.96 and 𝑧=1.96,
then we failed to reject the null hypothesis.
Try
Let’sit!Practice
Example 1: Construct the rejection region for a right-tailed test
at 0.05 level of significance.
Solution to Let’s Practice
Example 1: Construct the rejection region for a right-tailed test at 0.05 level of significance.
Solution:
A right-tailed test is a one-tailed test whose rejection region is to
the right side of the normal curve. Using the -table, the critical
value for a right-tailed test at 0.05 level of significance is .
Solution to Let’s Practice
Example 1: Construct the rejection region for a right-tailed test at 0.05 level of significance.
Solution:
The rejection region is illustrated below.
Try
Let’sit!Practice
Example 2: A claim about a population proportion is tested
using a left-tailed test at 0.05 level of significance. The test
statistic value is computed to be . Should the null hypothesis
be rejected or not?
Solution to Let’s Practice
Example 2: A claim about a population proportion is tested using a left-tailed test at 0.05 level
of significance. The test statistic value is computed to be . Should the null hypothesis be
rejected or not?
Solution:
1. Determine the critical value.
Since the level of significance is and the test is a left-tailed test,
the corresponding critical value is .
Solution to Let’s Practice
Example 2: A claim about a population proportion is tested using a left-tailed test at 0.05 level
of significance. The test statistic value is computed to be . Should the null hypothesis be
rejected or not?
Solution:
2. Illustrate the rejection region.
Since this is a left-tailed test, the rejection region is to the left
of the critical value .
Solution to Let’s Practice
Example 2: A claim about a population proportion is tested using a left-tailed test at 0.05 level
of significance. The test statistic value is computed to be . Should the null hypothesis be
rejected or not?
Solution:
2. Illustrate the rejection region.
Solution to Let’s Practice
Example 2: A claim about a population proportion is tested using a left-tailed test at 0.05 level
of significance. The test statistic value is computed to be . Should the null hypothesis be
rejected or not?
Solution:
3. Determine if the test statistic value falls in the rejection
region.
The test statistic value is . Locate this point on the normal
curve and check if it falls in the rejection region.
Solution to Let’s Practice
Example 2: A claim about a population proportion is tested using a left-tailed test at 0.05 level
of significance. The test statistic value is computed to be . Should the null hypothesis be
rejected or not?
Solution:
3. Determine if the test statistic value falls in the rejection
region.
Solution to Let’s Practice
Example 2: A claim about a population proportion is tested using a left-tailed test at 0.05 level
of significance. The test statistic value is computed to be . Should the null hypothesis be
rejected or not?
Solution:
4. Determine whether the null hypothesis should be rejected
or not.
Since the test statistic value falls in the rejection region, the
null hypothesis should be rejected.
Try It!
Individual Practice
1. Construct the rejection region for a two-tailed test
at 0.01 level of significance.
2. A claim about a population proportion is tested
using a two-tailed test at 0.05 level of significance.
The test statistic value is computed to be . Should
the null hypothesis be rejected or not?
Try It!
Group Practice: To be done in groups of three or four
Based on previous data, the company believes that at least
4% of the speakers they produce in the factory are defective.
However, the supervisor thinks that there are fewer
speakers that are defective now. This claim is tested at 0.05
level of significance, and the test statistic value is computed
to be . Write the null and alternative hypotheses, determine
whether the null hypothesis should be rejected, and
interpret the decision.
Key Points
● The null hypothesis is a statement which states that there
is no difference between a parameter and a specific value.
It is denoted by
● The alternative hypothesis is a statement which states
that there is a difference between a parameter and a
specific value. It is the negation of the null hypothesis.
Key Points
● A one-tailed test is a type of hypothesis test that makes
use of only one side or tail of the distribution. It can either
be a right-tailed or left-tailed test.
● A two-tailed test is a type of hypothesis test that makes
use of two opposite sides or tails of the distribution. It is
used if the alternative hypothesis contains the “not equal
to” symbol.
Key Points
● The level of significance is the probability of making an
error in rejecting the null hypothesis when it is actually
true.
● The critical values are -scores that are boundaries of the
rejection region.
Key Points
● When the test statistic value falls in the rejection region,
we reject the null hypothesis. Otherwise, we failed to
reject the null hypothesis.
Bibliography
Berman, Harvey. “Hypothesis Test for a Proportion.” Stat Trek. Retrieved 30 August 2019 from
https://bit.ly/2zsxnnr
Khan Academy. “Calculating a z statistic in a test about a proportion.” Retrieved 30 August 2019 from
https://bit.ly/2Lf2ZSN
The Pennsylvania State University. “Test of Proportion.” Statistics Online. Retrieved 30 August 2019 from
https://bit.ly/2HywduR
