**Statistics** This brand new series has been written for the University of Cambridge International Examinations course for AS and A Level Mathematics 9709 This title covers the requirements of S1 and S2 The authors are experienced examiners and teachers who have written extensively at this level so have ensured all mathematical concepts are explained using language and terminology that is appropriate for students across the world Students are provded with clear and detailed worked examples and questions

Publisher : Hodder Education

Author : Roger Porkess, Sophie Goldie

ISBN : 9781444146509

Year : 2012

Language: en

File Size : 5.02 MB

Category : Science Math

Cambridge

International AS and A Level Mathematics

Statistics

Sophie Goldie

Series Editor: Roger Porkess

Questions from the Cambridge International Examinations AS and A Level Mathematics papers

are reproduced by permission of University of Cambridge International Examinations.

Questions from the MEI AS and A Level Mathematics papers are reproduced by permission of OCR.

We are grateful to the following companies, institutions and individuals who have given permission

to reproduce photographs in this book.

Photo credits: page 3 © Artur Shevel / Fotolia; page 77 © Luminis / Fotolia; page 105 © Ivan Kuzmin / Alamy; page 123

© S. Ferguson; page 134 © Peter Küng / Fotolia; page 141 © Mathematics in Education and Industry; p.192 © Claudia

Paulussen / Fotolia.com; page 202 © Ingram Publishing Limited; page 210 © Peter Titmuss / Alamy; page 216 © Monkey

Business / Fotolia; page 233 © StockHouse / Fotolia; page 236 © Ingram Publishing Limited / Ingram Image Library

500-Animals; page 256 © Kevin Peterson / Photodisc / Getty Images; page 277 © Charlie Edwards / Getty Images;

page 285 © Stuart Miles / Fotolia.com

All designated trademarks and brands are protected by their respective trademarks.

Every effort has been made to trace and acknowledge ownership of copyright. The publishers will be

glad to make suitable arrangements with any copyright holders whom it has not been possible to contact.

Hachette UK’s policy is to use papers that are natural, renewable and recyclable products and

made from wood grown in sustainable forests. The logging and manufacturing processes are

expected to conform to the environmental regulations of the country of origin.

Orders: please contact Bookpoint Ltd, 130 Milton Park, Abingdon, Oxon OX14 4SB.

Telephone: (44) 01235 827720. Fax: (44) 01235 400454. Lines are open 9.00–5.00, Monday

to Saturday, with a 24-hour message answering service. Visit our website at www.hoddereducation.co.uk

Much of the material in this book was published originally as part of the MEI Structured

Mathematics series. It has been carefully adapted for the Cambridge International AS and A Level

Mathematics syllabus.

The original MEI author team for Statistics comprised Michael Davies, Ray Dunnett, Anthony Eccles,

Bob Francis, Bill Gibson, Gerald Goddall, Alan Graham, Nigel Green and Roger Porkess.

Copyright in this format © Roger Porkess and Sophie Goldie, 2012

First published in 2012 by

Hodder Education, an Hachette UK company,

338 Euston Road

London NW1 3BH

Impression number 5 4 3 2 1

Year

2016 2015 2014 2013 2012

All rights reserved. Apart from any use permitted under UK copyright law, no part of this

publication may be reproduced or transmitted in any form or by any means, electronic or

mechanical, including photocopying and recording, or held within any information storage

and retrieval system, without permission in writing from the publisher or under licence from

the Copyright Licensing Agency Limited. Further details of such licences (for reprographic

reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron

House, 6–10 Kirby Street, London EC1N 8TS.

Cover photo © Kaz Chiba/Photodisc/Getty Images/Natural Patterns BS13

Illustrations by Pantek Media, Maidstone, Kent

Typeset in 10.5pt Minion by Pantek Media, Maidstone, Kent

Printed in Dubai

A catalogue record for this title is available from the British Library

ISBN 978 1444 14650 9

Contents

Key to symbols in this book

vi

Introduction

vii

The Cambridge International AS and A Level Mathematics syllabus viii

S1 Statistics 1

1

Chapter 1

Exploring data

Looking at the data

Stem-and-leaf diagrams

Categorical or qualitative data

Numerical or quantitative data

Measures of central tendency

Frequency distributions

Grouped data

Measures of spread (variation)

Working with an assumed mean

2

4

7

13

13

14

19

24

34

45

Chapter 2

Representing and interpreting data

Histograms

Measures of central tendency and of spread using quartiles

Cumulative frequency curves

52

53

62

65

Chapter 3

Probability

Measuring probability

Estimating probability

Expectation

The probability of either one event or another

Independent and dependent events

Conditional probability

77

78

79

81

82

87

94

Chapter 4

Discrete random variables

Discrete random variables

Expectation and variance

105

106

114

iii

Chapter 5

Permutations and combinations

Factorials

Permutations

Combinations

The binomial coefficients

Using binomial coefficients to calculate probabilities

123

124

129

130

132

133

Chapter 6

The binomial distribution

The binomial distribution

The expectation and variance of B(n, p)

Using the binomial distribution

141

143

146

147

Chapter 7

The normal distribution

Using normal distribution tables

The normal curve

Modelling discrete situations

Using the normal distribution as an approximation for the

binomial distribution

154

156

161

172

S2 Statistics 2

iv

173

179

Chapter 8

Hypothesis testing using the binomial distribution

Defining terms

Hypothesis testing checklist

Choosing the significance level

Critical values and critical (rejection) regions

One-tail and two-tail tests

Type I and Type II errors

180

182

183

184

189

193

196

Chapter 9

The Poisson distribution

The Poisson distribution

Modelling with a Poisson distribution

The sum of two or more Poisson distributions

The Poisson approximation to the binomial distribution

Using the normal distribution as an approximation for the

Poisson distribution

202

204

207

210

216

224

Chapter 10

Continuous random variables

Probability density function

Mean and variance

The median

The mode

The uniform (rectangular) distribution

233

235

244

246

247

249

Chapter 11

Linear combinations of random variables

The expectation (mean) of a function of X, E(g[X])

Expectation: algebraic results

The sums and differences of independent random variables

More than two independent random variables

256

256

258

262

269

Chapter 12

Sampling

Terms and notation

Sampling

Sampling techniques

277

277

278

281

Chapter 13

Hypothesis testing and confidence intervals using

the normal distribution

Interpreting sample data using the normal distribution

The Central Limit Theorem

Confidence intervals

How large a sample do you need?

Confidence intervals for a proportion

285

285

298

300

304

306

Answers

Index

312

342

v

Key to symbols in this book

?

●

This symbol means that you may want to discuss a point with your teacher. If

you are working on your own there are answers in the back of the book. It is

important, however, that you have a go at answering the questions before looking

up the answers if you are to understand the mathematics fully.

! This is a warning sign. It is used where a common mistake, misunderstanding or

tricky point is being described.

This is the ICT icon. It indicates where you could use a graphic calculator or a

computer. Graphic calculators and computers are not permitted in any of the

examinations for the Cambridge International AS and A Level Mathematics 9709

syllabus, however, so these activities are optional.

This symbol and a dotted line down the right-hand side of the page indicate

material which is beyond the syllabus for the unit but which is included for

completeness.

vi

Introduction

This is part of a series of books for the University of Cambridge International

Examinations syllabus for Cambridge International AS and A Level Mathematics

9709. There are thirteen chapters in this book; the first seven cover Statistics 1

and the remaining six Statistics 2. The series also includes two books for pure

mathematics and one for mechanics.

These books are based on the highly successful series for the Mathematics in

Education and Industry (MEI) syllabus in the UK but they have been redesigned

for Cambridge international students; where appropriate, new material has been

written and the exercises contain many past Cambridge examination questions.

An overview of the units making up the Cambridge international syllabus is given

in the diagram on the next page.

Throughout the series the emphasis is on understanding the mathematics as well

as routine calculations. The various exercises provide plenty of scope for practising

basic techniques; they also contain many typical examination questions.

An important feature of this series is the electronic support. There is an

accompanying disc containing two types of Personal Tutor presentation:

examination-style questions, in which the solutions are written out, step by step,

with an accompanying verbal explanation, and test-yourself questions; these are

multiple-choice with explanations of the mistakes that lead to the wrong answers

as well as full solutions for the correct ones. In addition, extensive online support

is available via the MEI website, www.mei.org.uk.

The books are written on the assumption that students have covered and

understood the work in the Cambridge IGCSE® syllabus. However, some

of the early material is designed to provide an overlap and this is designated

‘Background’. There are also places where the books show how the ideas can be

taken further or where fundamental underpinning work is explored and such

work is marked as ‘Extension’.

The original MEI author team would like to thank Sophie Goldie who has carried

out the extensive task of presenting their work in a suitable form for Cambridge

international students and for her original contributions. They would also like to

thank University of Cambridge International Examinations for their detailed advice

in preparing the books and for permission to use many past examination questions.

Roger Porkess

Series Editor

vii

The Cambridge International AS

and A Level Mathematics syllabus

P2

Cambridge

IGCSE

Mathematics

P1

S1

AS Level

Mathematics

M1

S1

M1

S2

P3

M1

viii

S1

M2

A Level

Mathematics

Statistics 1

S1

Exploring data

S1

1

2

1

Exploring data

A judicious man looks at statistics, not to get knowledge but to save

himself from having ignorance foisted on him.

Carlyle

Source: The Times 2012

The cuttings on page 2 all appeared in one newspaper on one day. Some of them

give data as figures, others display them as diagrams.

How do you interpret this information? Which data do you take seriously and

which do you dismiss as being insignificant or even misleading?

Exploring data

To answer these questions fully you need to understand how data are collected

and analysed before they are presented to you, and how you should evaluate what

you are given to read (or see on the television). This is an important part of the

subject of statistics.

S1

1

In this book, many of the examples are set as stories from fictional websites.

Some of them are written as articles or blogs; others are presented from the

journalists’ viewpoint as they sort through data trying to write an interesting

story. As you work through the book, look too at the ways you are given such

information in your everyday life.

bikingtoday.com

Another cyclist seriously hurt. Will you be next?

On her way back home from school on

Wednesday afternoon, little Rita Roy

was knocked off her bicycle and taken to

hospital with suspected concussion.

Rita was struck by a Ford Transit van, only

50 metres from her own house.

Rita is the fourth child from the Nelson

Mandela estate to be involved in a serious

cycling accident this year.

The busy road where Rita Roy was

knocked off her bicycle yesterday.

After reading the blog, the editor of a local newspaper commissioned one of the

paper’s reporters to investigate the situation and write a leading article for the

paper on it. She explained to the reporter that there was growing concern locally

about cycling accidents involving children. She emphasised the need to collect

good quality data to support presentations to the paper’s readers.

?

●

Is the aim of the investigation clear?

Is the investigation worth carrying out?

What makes good quality data?

The reporter started by collecting data from two sources. He went through back

numbers of the newspaper for the previous two years, finding all the reports of

cycling accidents. He also asked an assistant to carry out a survey of the ages of

3

Exploring data

S1

1

local cyclists; he wanted to know whether most cyclists were children, young

adults or whatever.

?

●

Are the reporter’s data sources appropriate?

Before starting to write his article, the reporter needed to make sense of the data

for himself. He then had to decide how he was going to present the information

to his readers. These are the sorts of data he had to work with.

Name

Age

Distance

from home

Cause

Injuries

Treatment

Rahim Khan

45

3 km

skid

Concussion

Hospital

outpatient

Debbie Lane

5

75 km

hit kerb

Broken arm

Hospital

outpatient

Arvinder Sethi

12

1200 m

lorry

Multiple

fractures

Hospital

3 weeks

Husna Mahar

8

300 m

Bruising

Hospital

outpatient

David Huker

8

50 m

hit

each

other

Concussion

Hospital

outpatient

}

There were 92 accidents listed in the reporter’s table.

Ages of cyclists (from survey)

66 6 62 19 20

35 26 61 13 61

64 11 39 22 9

37 18 138 16 67

9 23 12 9 37

18 20 11 25 7

18 15

15

28

13

45

7

42

21

21

9

10

36

29

8 21 63

7 10 52

17 64 32

55 14 66

9 88 46

6 60 60

44 10 44 34 18

13 52 20 17 26

8 9 31 19 22

67 14 62 28 36

12 59 61 22 49

16 50 16 34 14

This information is described as raw data, which means that no attempt has yet

been made to organise it in order to look for any patterns.

Looking at the data

4

At the moment the arrangement of the ages of the 92 cyclists tells you very little

at all. Clearly these data must be organised so as to reveal the underlying shape,

the distribution. The figures need to be ranked according to size and preferably

grouped as well. The reporter had asked an assistant to collect the information

and this was the order in which she presented it.

Tally

Tallying is a quick, straightforward way of grouping data into suitable intervals.

You have probably met it already.

Tally

Frequency

13

10–19

26

20–29

16

30–39

10

40–49

6

50–59

5

60–69

0–9

70–79

80–89

Looking at the data

Stated age

(years)

S1

1

14

0

1

1

130–139

Total

92

Extreme values

A tally immediately shows up any extreme values, that is values which are far

away from the rest. In this case there are two extreme values, usually referred to

as outliers: 88 and 138. Before doing anything else you must investigate these.

In this case the 88 is genuine, the age of Millie Smith, who is a familiar sight

cycling to the shops.

The 138 needless to say is not genuine. It was the written response of a man who

was insulted at being asked his age. Since no other information about him is

available, this figure is best ignored and the sample size reduced from 92 to 91.

You should always try to understand an outlier before deciding to ignore it; it

may be giving you important information.

! Practical statisticians are frequently faced with the problem of outlying

observations, observations that depart in some way from the general pattern of

a data set. What they, and you, have to decide is whether any such observations

belong to the data set or not. In the above example the data value 88 is a genuine

member of the data set and is retained. The data value 138 is not a member of the

data set and is therefore rejected.

5

Describing the shape of a distribution

An obvious benefit of using a tally is that it shows the overall shape of the

distribution.

30

frequency density (people/10 years)

Exploring data

S1

1

20

10

0

10

20

30

40

50

60

70

80

90

age (years)

Figure 1.1 Histogram to show the ages of people involved in cycling accidents

You can now see that a large proportion (more than a quarter) of the sample are

in the 10 to 19 year age range. This is the modal group as it is the one with the

most members. The single value with the most members is called the mode, in

this case age 9.

You will also see that there is a second peak among those in their sixties; so this

distribution is called bimodal, even though the frequency in the interval 10–19 is

greater than the frequency in the interval 60–69.

Different types of distribution are described in terms of the position of their

modes or modal groups, see figure 1.2.

(a)

(b)

(c)

Figure 1.2 Distribution shapes: (a) unimodal and symmetrical (b) uniform (no

mode but symmetrical) (c) bimodal

6

When the mode is off to one side the distribution is said to be skewed. If the

mode is to the left with a long tail to the right the distribution has positive (or

right) skewness; if the long tail is to the left the distribution has negative (or left)

skewness. These two cases are shown in figure 1.3.

© 2018-2019 uberlabel.com. All rights reserved