Cambridge International AS and A Level ... - GCE Guide

1 Sophie GoldieSeries Editor: Roger PorkessCambridgeInternational AS and A Level MathematicsStatisticsQuestions from the Cambridge International Examinations AS and A Level Mathematics papers are reproduced by permission of university of Cambridge International 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 K ng / Fotolia; page 141 Mathematics in Education and Industry; Claudia Paulussen / ; 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 / designated trademarks and brands are protected by their respective effort has been made to trace and acknowledge ownership of copyright.

2 The publishers will be glad to make suitable arrangements with any copyright holders whom it has not been possible to 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 : please contact Bookpoint Ltd, 130 Milton Park, Abingdon, Oxon OX14 : (44) 01235 827720. Fax: (44) 01235 400454. Lines are open , Monday to Saturday, with a 24-hour message answering service. Visit our website at of the material in this book was published originally as part of the MEI Structured Mathematics series.

3 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 in this format Roger Porkess and Sophie Goldie, 2012 First published in 2012 byHodder Education, an Hachette UK company,338 Euston RoadLondon NW1 3 BHImpression 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.

4 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 photo Kaz Chiba/Photodisc/Getty Images/Natural Patterns BS13 Illustrations by Pantek Media, Maidstone, KentTypeset in Minion by Pantek Media, Maidstone, KentPrinted in DubaiA catalogue record for this title is available from the British LibraryISBN 978 1444 14650 9 ContentsKey to symbols in this book viIntroduction viiThe Cambridge International AS and A Level Mathematics syllabus viiiS1 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)

5 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 105 Discrete random variables 106 Expectation and variance 114 Chapter 1 Chapter 2 Chapter 3 Chapter 4iiiContentsKey to symbols in this book viIntroduction viiThe Cambridge International AS and A Level Mathematics syllabus viiiS1 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)

6 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 105 Discrete random variables 106 Expectation and variance 114 Chapter 1 Chapter 2 Chapter 3 Chapter 4ivContinuous random variables 233 Probability density function 235 Mean and variance 244 The median 246 The mode 247 The uniform (rectangular) distribution 249 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 Sampling 277 Terms and notation 277 Sampling 278 Sampling techniques 281 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?

7 304 Confidence intervals for a proportion 306 Answers 312 Index 342 Chapter 10 Chapter 11 Chapter 12 Permutations and combinations 123 Factorials 124 Permutations 129 Combinations 130 The binomial coefficients 132 Using binomial coefficients to calculate probabilities 133 The binomial distribution 141 The binomial distribution 143 The expectation and variance of B(n, p) 146 Using the binomial distribution 147 The normal distribution 154 Using normal distribution tables 156 The normal curve 161 Modelling discrete situations 172 Using the normal distribution as an approximation for the binomial distribution 173S2 Statistics 2 179 Hypothesis testing using the binomial distribution 180 Defining terms 182 Hypothesis testing checklist 183 Choosing the significance Level 184 Critical values and critical (rejection)

8 Regions 189 One-tail and two-tail tests 193 Type I and Type II errors 196 The Poisson distribution 202 The Poisson distribution 204 Modelling with a Poisson distribution 207 The sum of two or more Poisson distributions 210 The Poisson approximation to the binomial distribution 216 Using the normal distribution as an approximation for the Poisson distribution 224 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9vvContinuous random variables 233 Probability density function 235 Mean and variance 244 The median 246 The mode 247 The uniform (rectangular) distribution 249 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 Sampling 277 Terms and notation 277 Sampling 278 Sampling techniques 281 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?

9 304 Confidence intervals for a proportion 306 Answers 312 Index 342 Chapter 10 Chapter 11 Chapter 12 Chapter 13 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 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 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 is part of a series of books for the university of Cambridge International Examinations syllabus for Cambridge International AS and A Level Mathematics 9709.

10 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 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 the series the emphasis is on understanding the mathematics as well as routine calculations.