Cambridge University Press
978-1-316-60038-2 — Cambridge International AS and A Level Mathematics: Revised Edition Statistics 1 Coursebook
Steve Dobbs , Jane Miller , Julian Gilbey
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Steve Dobbs, Jane Miller and Julian Gilbey
Cambridge International
AS and A Level Mathematics:
Statistics 1
Coursebook
Revised Edition
© in this web service Cambridge University Press www.cambridge.org
Cambridge University Press
978-1-316-60038-2 — Cambridge International AS and A Level Mathematics: Revised Edition Statistics 1 Coursebook
Steve Dobbs , Jane Miller , Julian Gilbey
Frontmatter
More Information
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First published 2002
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Image credits: Cover Georgette Douwma/Science Photo Library; Chapter 1
Eshkin/Shutterstock; Chapter 2 echo3005/Shutterstock; Chapter 3 Sergey Nivens/
Shutterstock; Chapter 4 ezraff yusoff/Shutterstock; Chapter 5 Marian Mais/
Shutterstock; Chapter 6 ktsdesign/Shutterstock; Chapters 7, 9 Ken Schulze/
Shutterstock; Chapter 8 DanielAngmel/Shutterstock
© in this web service Cambridge University Press www.cambridge.org
Cambridge University Press
978-1-316-60038-2 — Cambridge International AS and A Level Mathematics: Revised Edition Statistics 1 Coursebook
Steve Dobbs , Jane Miller , Julian Gilbey
Frontmatter
More Information
Contents
Introduction vi
1 Representation of data 1
1.1 Introduction 2
1.2 Stem-and-leaf diagrams 5
1.3 Histograms 10
1.4 Cumulative frequency graphs 19
1.5 Practical activities 21
2 Measures of location 25
2.1 Introduction 26
2.2 The median 26
2.3 Finding the median from a frequency table 27
2.4 The mean 29
2.5 Summation notation 29
2.6 Calculating the mean from a frequency table 30
2.7 Making the calculation of the mean easier 32 iii
2.8 The mode and the modal class 34
2.9 Comparison of the mean, median and mode 35
2.10 Practical activities 37
3 Measures of spread 41
3.1 Introduction 42
3.2 The range 42
3.3 The interquartile range 43
3.4 The ive-number summary 47
3.5 Box-and-whisker plots 48
3.6 Outliers 50
3.7 Variance and standard deviation 52
3.8 Proof of the equivalence of the variance formulae 55
3.9 Calculating variance from a frequency table 56
3.10 Making the calculation of variance easier 58
3.11 Choosing how to represent data 61
3.12 Practical activities 63
© in this web service Cambridge University Press www.cambridge.org
Cambridge University Press
978-1-316-60038-2 — Cambridge International AS and A Level Mathematics: Revised Edition Statistics 1 Coursebook
Steve Dobbs , Jane Miller , Julian Gilbey
Frontmatter
More Information
Contents
4 Probability 71
4.1 Assigning probability 72
4.2 Probabilities of events 73
4.3 Addition of probabilities 76
4.4 Conditional probability 78
4.5 Independent events 82
4.6 Practical activities 85
5 Permutations and combinations 92
5.1 Permutations 93
5.2 Permutations when the objects are not distinct 96
5.3 Combinations 98
5.4 Applications of permutations and combinations 101
6 Probability distributions 111
6.1 Discrete random variables 112
6.2 An important property of a probability distribution 116
6.3 Using a probability distribution as a model 119
iv 7 The binomial distribution 122
7.1 The binomial distribution 123
7.2 Using the binomial distribution as a model 128
7.3 Practical activities 130
8 Expectation and variance of a random variable 134
8.1 Expectation 135
8.2 The variance of a random variable 137
8.3 The expectation and variance of a binomial distribution 141
9 The normal distribution 148
9.1 Modelling continuous variables 149
9.2 The normal distribution 152
9.3 The standard normal distribution 154
9.4 Standardising a normal distribution 159
9.5 Modelling with the normal distribution 162
9.6 Practical activities 164
9.7 The normal distribution as an approximation to the
binomial distribution 166
© in this web service Cambridge University Press www.cambridge.org
Cambridge University Press
978-1-316-60038-2 — Cambridge International AS and A Level Mathematics: Revised Edition Statistics 1 Coursebook
Steve Dobbs , Jane Miller , Julian Gilbey
Frontmatter
More Information
Contents
Revision exercise 177
Practice exam-style papers 181
The normal distribution function 185
Answers 187
Index 200
© in this web service Cambridge University Press www.cambridge.org
Cambridge University Press
978-1-316-60038-2 — Cambridge International AS and A Level Mathematics: Revised Edition Statistics 1 Coursebook
Steve Dobbs , Jane Miller , Julian Gilbey
Frontmatter
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Introduction
Cambridge International AS and A Level Mathematics has been written especially for
the Cambridge International Examinations syllabus 9709. There is one book
corresponding to each syllabus unit, except that units P2 and P3 are contained in a
single book. This book is the irst Probability and Statistics unit, S1.
The syllabus content is arranged by chapters which are ordered so as to provide a
viable teaching course. A few sections include important results that are dificult to
prove or outside the syllabus. These sections are marked with a vertical coloured bar
in the section heading, and there is usually a sentence early on explaining precisely
what it is that the student needs to know.
Some paragraphs within the text appear in this type style. These paragraphs are usually
outside the main stream of the mathematical argument, but may help to give insight,
or suggest extra work or different approaches.
Graphic calculators are not permitted in the examination, but they can be useful aids
in learning mathematics. In the book the authors have noted where access to graphic
calculators would be especially helpful but have not assumed that they are available
to all students.
The authors have assumed that students have access to calculators with built-in
statistical functions.
Numerical work is presented in a form intended to discourage premature
vi
approximation.
In ongoing calculations inexact numbers appear in decimal form like 3.456...,
signifying that the number is held in a calculator to more places than are given.
Numbers are not rounded at this stage; the full display could be either 3.456 123
or 3.456 789. Final answers are then stated with some indication that they are
approximate, for example ‘1.23 correct to 3 signiicant igures’.
Most chapters contain Practical activities. These can be used either as an
introduction to a topic, or, later on, to reinforce the theory. There are also plenty of
exercises, and each chapter ends with a Miscellaneous exercise which includes some
questions of examination standard. There is a Revision exercise, and two Practice
examination-style papers.
In some exercises a few of the later questions may go beyond the likely requirements
of the examination, either in dificulty or in length or both. Some questions are
marked with a vertical coloured bar which indicates that they require knowledge of
results outside the syllabus.
Cambridge University Press would like to thank Cambridge International
Examinations for permission to use past paper examination questions.
The authors thank Cambridge International Examinations and Cambridge University
Press for their help in producing this book. However, the responsibility for the text,
and for any errors, remains with the authors.
Steve Dobbs and Jane Miller, 2002
© in this web service Cambridge University Press www.cambridge.org
Cambridge University Press
978-1-316-60038-2 — Cambridge International AS and A Level Mathematics: Revised Edition Statistics 1 Coursebook
Steve Dobbs , Jane Miller , Julian Gilbey
Frontmatter
More Information
Introduction Introduction
Introduction to the revised edition
This revised edition has been prepared to bring this textbook in line with the current
version of the Cambridge International Examinations speciication. Most of the
original edition has been left unchanged to assist teachers familiar with the original
edition; this includes section numbers, question numbers and so on. New text has
been added in Section 1.2 discussing back-to-back stem-and-leaf diagrams. A question
has been added to Exercise 1A, and two questions have been modiied in Chapter 3
to relect this. Two questions have been added to Exercise 3C on using coding to
calculate variance. In Chapter 4, it is noted that ‘exclusive events’ is an alternative to
‘mutually exclusive events’, as this is the nomenclature favoured by the examination
board. Also in Chapter 4, Section 4.5 on independent events has been enlarged
slightly, with a note that two events can be identiied as independent if
P(A and B) = P(A) × P(B) and two new examples. In Chapter 5, more modern
notation for the number of permutations has been introduced. Two new illustrative
examples, 5.4.5 and 6.2.3, have been added in light of several recent examination
questions on this theme.
The other major change in this edition is the replacement of all of the older OCR
examination questions in the exercises by more recent Cambridge International
Examinations questions. This will be of beneit to students preparing for the current
style of examination questions. In order to maintain the numbering of the other
questions, the newer questions have been slotted in to the exercises. While this has
inevitably meant some loss of order within the miscellaneous exercises, it was felt
to be more than compensated for by the preservation of the original numbering. vii
In a few of the exercises, an insuficient number of past paper questions were
available to replace the existing questions; in these cases, the exercises have been
shortened. In Chapters 3 and 8, there are many recent examination questions and
so the miscellaneous exercises have been extended to include illustrative examples.
Further past papers can, of course, be requested from Cambridge International
Examinations. All questions and answers taken from Cambridge International
Examinations past papers have been clearly referenced. All other questions and
answers have been written by the authors of this book.
The editor of this edition thanks Cambridge International Examinations and
Cambridge University Press, in particular Cathryn Freear and Andrew Briggs, for
their great help in preparing this revised edition.
Julian Gilbey
London, 2016
© in this web service Cambridge University Press www.cambridge.org
