Cambridge International AS and A Level Mathematics Statistics

[Pages:353]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.

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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

Chapter 1

Chapter 2 Chapter 3 Chapter 4

Key to symbols in this book

vi

Introduction

vii

The Cambridge International AS and A Level Mathematics syllabus viii

S1 Statistics 1

1

Exploring data

2

Looking at the data

4

Stem-and-leaf diagrams

7

Categorical or qualitative data

13

Numerical or quantitative data

13

Measures of central tendency

14

Frequency distributions

19

Grouped data

24

Measures of spread (variation)

34

Working with an assumed mean

45

Representing and interpreting data

52

Histograms

53

Measures of central tendency and of spread using quartiles

62

Cumulative frequency curves

65

Probability

77

Measuring probability

78

Estimating probability

79

Expectation

81

The probability of either one event or another

82

Independent and dependent events

87

Conditional probability

94

Discrete random variables Discrete random variables Expectation and variance

105 106 114

iii

Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9

Permutations and combinations Factorials Permutations Combinations The binomial coefficients Using binomial coefficients to calculate probabilities

The binomial distribution The binomial distribution The expectation and variance of B(n, p) Using the binomial distribution

The normal distribution Using normal distribution tables The normal curve Modelling discrete situations Using the normal distribution as an approximation for the binomial distribution

S2 Statistics 2

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

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

123 124 129 130 132 133

141 143 146 147

154 156 161 172

173

179

180 182 183 184 189 193 196

202 204 207 210 216

224

iv

Chapter 10

Continuous random variables

233

Probability density function

235

Mean and variance

244

The median

246

The mode

247

The uniform (rectangular) distribution

249

Chapter 11

Linear combinations of random variables

256

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

256

Expectation: algebraic results

258

The sums and differences of independent random variables

262

More than two independent random variables

269

Chapter 12

Sampling

277

Terms and notation

277

Sampling

278

Sampling techniques

281

Chapter 13

Hypothesis testing and confidence intervals using

the normal distribution

285

Interpreting sample data using the normal distribution

285

The Central Limit Theorem

298

Confidence intervals

300

How large a sample do you need?

304

Confidence intervals for a proportion

306

Answers

312

Index

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, .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

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