‘The Barrow exercises and online resources offer good scope for directing students to a great source of self study.’ Do you need to brush up on your statistical skills to truly excel in your economics or business course? If you want to increase your confidence in statistics then this is the perfect book for you The fifth edition of Statistics for Economics, Accounting and Business Studies continues to present a user-friendly and concise introduction to a variety of statistical tools and techniques Throughout the text, the author demonstrates how and why these techniques can be used to solve real-life problems, highlighting common mistakes and assuming no prior knowledge of the subject New to this fifth edition: • Chapter 11, Seasonal adjustment of time-series data is back by popular demand • New worked examples in every chapter and more real-life business examples – such as whether the level of general corruption in a country harms investment and whether boys or girls perform better at school – show how to apply an understanding of statistical techniques to wider business practice • New interactive online resource MathXL for Statistics See below for more details MathXL for Statistics A brand new online learning resource for this edition available to users of this book at www.pearsoned.co.uk/barrow This core textbook is aimed at undergraduate and MBA students taking an introductory statistics course on their economics, accounting or business studies degree • Interactive questions with randomised values allow you to practise the same concept as many times as you need until you master it • Guided solutions break down the question for you step-by-step • Audio animations talk you through key statistical techniques an imprint of CVR_BARR7942_05_SE_CVR.indd Michael Barrow is a Senior Lecturer in Economics at the University of Sussex He has acted as a consultant for major industrial, commercial and government bodies Front cover image: © Getty Images MICHAEL BARROW STATISTICS FOR ECONOMICS, ACCOUNTING AND BUSINESS STUDIES Fifth Edition BARROW An unrivalled online study and testing resource that generates a personalised study plan and provides extensive practice questions exactly where you need them STATISTICS FOR ECONOMICS, ‘There are thousands of intro stats books on the market, but few which are sufficiently orientated towards economics, and even fewer that treat topics with as much rigour as Barrow does.’ Andy Dickerson, University of Sheffield Fifth Edition ACCOUNTING AND BUSINESS STUDIES ‘An excellent reference book for the undergraduate student; filled with examples and applications – both practical (i.e computer-based) and traditional (i.e pen and paper problems); wide-ranging and sensibly ordered The book is written clearly, easy to follow yet not in the least patronising This is a particular strength.’ Christopher Gerry, UCL www.pearson-books.com 9/3/09 10:56:41 STFE_A01.qxd 26/02/2009 09:01 Page i Statistics for Economics, Accounting and Business Studies The Power of Practice With your purchase of a new copy of this textbook, you received a Student Access Kit for getting started with statistics using MathXL Follow the instructions on the card to register successfully and start making the most of the resources Don’t throw it away! The Power of Practice MathXL is an online study and testing resource that puts you in control of your study, providing extensive practice exactly where and when you need it MathXL gives you unrivalled resources: ● Sample tests for each chapter to see how much you have learned and where you still need practice ● A personalised study plan, which constantly adapts to your strengths and weaknesses, taking you to exercises you can practise over and over with different variables every time ● ‘Help me solve this’ provide guided solutions which break the problem into its component steps and guide you through with hints ● Audio animations guide you step-by-step through the key statistical techniques ● Click on the E-book textbook icon to read the relevant part of your textbook again See pages xiv–xv for more details To activate your registration go to www.pearsoned.co.uk/barrow and follow the instructions on-screen to register as a new user ➔ STFE_A01.qxd 26/02/2009 09:01 Page ii We work with leading authors to develop the strongest educational materials in Accounting, bringing cutting-edge thinking and best learning practice to a global market Under a range of well-known imprints, including Financial Times Prentice Hall, we craft high-quality print and electronic publications, which help readers to understand and apply their content, whether studying or at work To find out more about the complete range of our publishing, please visit us on the World Wide Web at: www.pearsoned.co.uk STFE_A01.qxd 26/02/2009 09:01 Page iii Statistics for Economics, Accounting and Business Studies Fifth Edition Michael Barrow University of Sussex STFE_A01.qxd 26/02/2009 09:01 Page iv Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk First published 1988 Fifth edition published 2009 © Pearson Education Limited 1988, 2009 The right of Michael Barrow to be identified as author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners ISBN 13: 978-0-273-71794-2 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Barrow, Michael Statistics for economics, accounting and business studies / Michael Barrow – 5th ed p com Includes bibliographical references and index ISBN 978-0-273-71794-2 (pbk : alk paper) Economics–Statistical methods Commercial statistics I Title HB137.B37 2009 519.5024′33–dc22 2009003125 10 13 12 11 10 09 Typeset in 9/12pt Stone Serif by 35 Printed and bound by Ashford Colour Press Ltd Gosport The publisher’s policy is to use paper manufactured from sustainable forests STFE_A01.qxd 26/02/2009 09:01 Page v For Patricia, Caroline and Nicolas STFE_A01.qxd 26/02/2009 09:01 Page vii Contents Guided tour of the book xii Getting started with statistics using MathXL xiv Preface to the fifth edition xvii Introduction Descriptive statistics Learning outcomes Introduction Summarising data using graphical techniques Looking at cross-section data: wealth in the UK in 2003 Summarising data using numerical techniques The box and whiskers diagram Time-series data: investment expenditures 1973–2005 Graphing bivariate data: the scatter diagram Data transformations Guidance to the student: how to measure your progress Summary Key terms and concepts Reference Problems Answers to exercises Appendix 1A: Σ notation Problems on Σ notation Appendix 1B: E and V operators Appendix 1C: Using logarithms Problems on logarithms Probability Learning outcomes Probability theory and statistical inference The definition of probability Probability theory: the building blocks Bayes’ theorem Decision analysis Summary Key terms and concepts Problems Answers to exercises 8 10 16 24 44 45 58 60 62 63 64 64 65 71 75 76 77 78 79 80 80 81 81 84 91 93 98 98 99 105 vii STFE_A01.qxd 26/02/2009 09:01 Page viii Contents Probability distributions Learning outcomes Introduction Random variables The Binomial distribution The Normal distribution The sample mean as a Normally distributed variable The relationship between the Binomial and Normal distributions The Poisson distribution Summary Key terms and concepts Problems Answers to exercises Estimation and confidence intervals Learning outcomes Introduction Point and interval estimation Rules and criteria for finding estimates Estimation with large samples Precisely what is a confidence interval? Estimation with small samples: the t distribution Summary Key terms and concepts Problems Answers to exercises Appendix: Derivations of sampling distributions Hypothesis testing Learning outcomes Introduction The concepts of hypothesis testing The Prob-value approach Significance, effect size and power Further hypothesis tests Hypothesis tests with small samples Are the test procedures valid? Hypothesis tests and confidence intervals Independent and dependent samples Discussion of hypothesis testing Summary Key terms and concepts Reference viii 108 108 109 110 111 117 125 131 132 135 136 137 142 144 144 145 145 146 149 153 160 165 165 166 169 170 172 172 173 173 180 181 183 187 189 190 191 194 195 196 196 STFE_A01.qxd 26/02/2009 09:01 Page ix Contents Problems Answers to exercises The χ and F distributions Learning outcomes Introduction The χ distribution The F distribution Analysis of variance Summary Key terms and concepts Problems Answers to exercises Appendix: Use of χ and F distribution tables Correlation and regression Learning outcomes Introduction What determines the birth rate in developing countries? Correlation Regression analysis Inference in the regression model Summary Key terms and concepts References Problems Answers to exercises Multiple regression Learning outcomes Introduction Principles of multiple regression What determines imports into the UK? Finding the right model Summary Key terms and concepts Reference Problems Answers to exercises Data collection and sampling methods Learning outcomes Introduction Using secondary data sources Using electronic sources of data 197 201 204 204 205 205 220 222 229 230 231 234 236 237 237 238 238 240 251 257 271 272 272 273 276 279 279 280 281 282 300 307 308 308 309 313 318 318 319 319 321 ix STFE_A01.qxd 26/02/2009 09:01 Page x Contents Collecting primary data The meaning of random sampling Calculating the required sample size Collecting the sample Case study: the UK Expenditure and Food Survey Summary Key terms and concepts References Problems 10 Index numbers Learning outcomes Introduction A simple index number A price index with more than one commodity Using expenditures as weights Quantity and expenditure indices The Retail Price Index Inequality indices The Lorenz curve The Gini coefficient Concentration ratios Summary Key terms and concepts References Problems Answers to exercises Appendix: Deriving the expenditure share form of the Laspeyres price index 11 Seasonal adjustment of time-series data 342 343 343 344 345 353 355 360 366 367 370 374 376 376 376 377 382 385 386 Learning outcomes Introduction The components of a time series Forecasting Further issues Summary Key terms and concepts Problems Answers to exercises 386 387 387 399 400 401 401 402 404 Important formulae used in this book 408 Appendix: Tables 412 412 414 Table A1 Table A2 x 323 324 333 335 338 339 340 340 341 Random number table The standard Normal distribution STFE_Z02.qxd 26/02/2009 09:20 Page 413 Appendix: Tables Table A1 Continued 78 67 93 35 32 92 40 57 82 100 30 10 50 68 52 100 73 41 41 11 30 37 83 98 29 75 28 64 67 21 78 14 76 18 78 77 50 79 44 41 62 42 67 68 26 50 78 30 43 59 86 22 96 39 73 35 43 51 79 85 92 95 94 61 92 12 15 76 77 41 34 44 33 32 60 72 89 21 16 27 19 38 88 82 25 98 89 74 73 27 95 66 23 48 90 76 40 31 21 27 77 78 48 59 85 70 90 85 76 30 82 22 31 30 30 25 66 53 25 26 82 46 19 83 40 20 76 67 52 57 92 89 64 29 11 32 63 90 28 68 43 97 88 48 78 94 43 41 45 94 21 63 91 91 97 64 45 21 85 59 62 42 27 39 89 34 65 64 20 27 36 57 61 12 32 14 69 67 39 99 86 28 80 32 93 32 13 57 55 36 35 80 78 47 42 81 96 31 57 37 66 99 35 82 39 58 11 90 73 39 35 55 47 36 21 35 93 71 52 63 87 38 83 24 21 92 88 84 25 21 22 25 12 13 65 15 20 69 87 66 56 97 17 70 70 79 99 70 76 34 53 10 54 32 30 76 12 91 85 18 84 90 37 19 81 10 22 47 43 85 68 92 28 85 61 56 26 51 66 56 81 59 39 95 59 46 24 23 60 29 86 49 70 56 67 37 66 21 23 84 70 80 66 84 95 45 37 14 37 27 13 51 26 28 76 62 62 92 11 32 16 47 57 48 29 85 92 48 84 38 33 43 44 19 93 81 74 82 55 51 18 39 31 16 63 27 51 27 30 39 24 81 51 48 55 78 77 45 18 19 92 70 12 98 83 40 80 12 99 63 26 35 71 42 99 50 37 65 10 60 95 22 31 13 20 100 93 65 32 66 97 66 17 89 16 82 78 64 72 52 78 63 48 67 79 55 64 78 25 56 87 22 53 96 66 79 75 27 50 60 14 65 63 34 34 88 10 32 33 51 28 73 40 19 86 22 76 413 STFE_Z02.qxd 26/02/2009 09:20 Page 414 Appendix: Tables Table A2 The standard Normal distribution z 414 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.1 0.2 0.3 0.4 0.5000 0.4602 0.4207 0.3821 0.3446 0.4960 0.4562 0.4168 0.3783 0.3409 0.4920 0.4522 0.4129 0.3745 0.3372 0.4880 0.4483 0.4090 0.3707 0.3336 0.4840 0.4443 0.4052 0.3669 0.3300 0.4801 0.4404 0.4013 0.3632 0.3264 0.4761 0.4364 0.3974 0.3594 0.3228 0.4721 0.4325 0.3936 0.3557 0.3192 0.4681 0.4286 0.3897 0.3520 0.3156 0.4641 0.4247 0.3859 0.3483 0.3121 0.5 0.6 0.7 0.8 0.9 0.3085 0.2743 0.2420 0.2119 0.1841 0.3050 0.2709 0.2389 0.2090 0.1814 0.3015 0.2676 0.2358 0.2061 0.1788 0.2981 0.2643 0.2327 0.2033 0.1762 0.2946 0.2611 0.2296 0.2005 0.1736 0.2912 0.2578 0.2266 0.1977 0.1711 0.2877 0.2546 0.2236 0.1949 0.1685 0.2843 0.2514 0.2206 0.1922 0.1660 0.2810 0.2483 0.2177 0.1894 0.1635 0.2776 0.2451 0.2148 0.1867 0.1611 1.0 1.1 1.2 1.3 1.4 0.1587 0.1357 0.1151 0.0968 0.0808 0.1562 0.1335 0.1131 0.0951 0.0793 0.1539 0.1314 0.1112 0.0934 0.0778 0.1515 0.1292 0.1093 0.0918 0.0764 0.1492 0.1271 0.1075 0.0901 0.0749 0.1469 0.1251 0.1056 0.0885 0.0735 0.1446 0.1230 0.1038 0.0869 0.0721 0.1423 0.1210 0.1020 0.0853 0.0708 0.1401 0.1190 0.1003 0.0838 0.0694 0.1379 0.1170 0.0985 0.0823 0.0681 1.5 1.6 1.7 1.8 1.9 0.0668 0.0548 0.0446 0.0359 0.0287 0.0655 0.0537 0.0436 0.0351 0.0281 0.0643 0.0526 0.0427 0.0344 0.0274 0.0630 0.0516 0.0418 0.0336 0.0268 0.0618 0.0505 0.0409 0.0329 0.0262 0.0606 0.0495 0.0401 0.0322 0.0256 0.0594 0.0485 0.0392 0.0314 0.0250 0.0582 0.0475 0.0384 0.0307 0.0244 0.0571 0.0465 0.0375 0.0301 0.0239 0.0559 0.0455 0.0367 0.0294 0.0233 2.0 2.1 2.2 2.3 2.4 0.0228 0.0179 0.0139 0.0107 0.0082 0.0222 0.0174 0.0136 0.0104 0.0080 0.0217 0.0170 0.0132 0.0102 0.0078 0.0212 0.0166 0.0129 0.0099 0.0075 0.0207 0.0162 0.0125 0.0096 0.0073 0.0202 0.0158 0.0122 0.0094 0.0071 0.0197 0.0154 0.0119 0.0091 0.0069 0.0192 0.0150 0.0116 0.0089 0.0068 0.0188 0.0146 0.0113 0.0087 0.0066 0.0183 0.0143 0.0110 0.0084 0.0064 2.5 2.6 2.7 2.8 2.9 3.0 0.0062 0.0047 0.0035 0.0026 0.0019 0.0013 0.0060 0.0045 0.0034 0.0025 0.0018 0.0013 0.0059 0.0044 0.0033 0.0024 0.0018 0.0013 0.0057 0.0043 0.0032 0.0023 0.0017 0.0012 0.0055 0.0041 0.0031 0.0023 0.0016 0.0012 0.0054 0.0040 0.0030 0.0022 0.0016 0.0011 0.0052 0.0039 0.0029 0.0021 0.0015 0.0011 0.0051 0.0038 0.0028 0.0021 0.0015 0.0011 0.0049 0.0037 0.0027 0.0020 0.0014 0.0010 0.0048 0.0036 0.0026 0.0019 0.0014 0.0010 STFE_Z02.qxd 26/02/2009 09:20 Page 415 Appendix: Tables Table A3 Percentage points of the t distribution The table gives critical values of the t distribution cutting off an area α in each tail, shown by the top row of the table Area (α) in each tail v 0.4 0.25 0.1 0.05 0.025 0.01 0.005 0.0025 0.001 0.0005 0.325 0.289 0.277 0.271 0.267 1.000 0.816 0.765 0.741 0.727 3.078 1.886 1.638 1.533 1.476 6.314 2.920 2.353 2.132 2.015 12.706 4.303 3.182 2.776 2.571 31.821 6.965 4.541 3.747 3.365 63.657 9.925 5.841 4.604 4.032 127.320 14.089 7.453 5.598 4.773 318.310 22.327 10.214 7.173 5.893 636.620 31.598 12.924 8.610 6.869 10 0.265 0.263 0.262 0.261 0.260 0.718 0.711 0.706 0.703 0.700 1.440 1.415 1.397 1.383 1.372 1.943 1.895 1.860 1.833 1.812 2.447 2.365 2.306 2.262 2.228 3.143 2.998 2.896 2.821 2.764 3.707 3.499 3.355 3.250 3.169 4.317 4.029 3.833 3.690 3.581 5.208 4.785 4.501 4.297 4.144 5.959 5.408 5.041 4.781 4.587 11 12 13 14 15 0.260 0.259 0.259 0.258 0.258 0.697 0.695 0.694 0.692 0.691 1.363 1.356 1.350 1.345 1.341 1.796 1.782 1.771 1.761 1.753 2.201 2.179 2.160 2.145 2.131 2.718 2.681 2.650 2.624 2.602 3.106 3.055 3.012 2.977 2.947 3.497 3.428 3.372 3.326 3.286 4.025 3.930 3.852 3.787 3.733 4.437 4.318 4.221 4.140 4.073 16 17 18 19 20 0.258 0.257 0.257 0.257 0.257 0.690 0.689 0.688 0.688 0.687 1.337 1.333 1.330 1.328 1.325 1.746 1.740 1.734 1.729 1.725 2.120 2.110 2.101 2.093 2.086 2.583 2.567 2.552 2.539 2.528 2.921 2.898 2.878 2.861 2.845 3.252 3.222 3.197 3.174 3.153 3.686 3.646 3.610 3.579 3.552 4.015 3.965 3.922 3.883 3.850 21 22 23 24 25 0.257 0.256 0.256 0.256 0.256 0.686 0.686 0.685 0.685 0.684 1.323 1.321 1.319 1.318 1.316 1.721 1.717 1.714 1.711 1.708 2.080 2.074 2.069 2.064 2.060 2.518 2.508 2.500 2.492 2.485 2.831 2.819 2.807 2.797 2.787 3.135 3.119 3.104 3.091 3.078 3.527 3.505 3.485 3.467 3.450 3.819 3.792 3.767 3.745 3.725 26 27 28 29 30 0.256 0.256 0.256 0.256 0.256 0.684 0.684 0.683 0.683 0.683 1.315 1.314 1.313 1.311 1.310 1.706 1.703 1.701 1.699 1.697 2.056 2.052 2.048 2.045 2.042 2.479 2.473 2.467 2.462 2.457 2.779 2.771 2.763 2.756 2.750 3.067 3.057 3.047 3.038 3.030 3.435 3.421 3.408 3.396 3.385 3.707 3.690 3.674 3.659 3.646 40 60 120 ∞ 0.255 0.254 0.254 0.253 0.681 0.679 0.677 0.674 1.303 1.296 1.289 1.282 1.684 1.671 1.658 1.645 2.021 2.000 1.980 1.960 2.423 2.390 2.358 2.326 2.704 2.660 2.617 2.576 2.971 2.915 2.860 2.807 3.307 3.232 3.160 3.090 3.551 3.460 3.373 3.291 415 STFE_Z02.qxd 26/02/2009 09:20 Page 416 Appendix: Tables Table A4 Critical values of the χ2 distribution The values in the table give the critical values of χ2 which cut off the area in the right-hand tail given at the top of the column Area in right-hand tail v 0.995 0.900 0.750 0.500 −10 392 704.10 0.0100251 0.0717218 0.206989 0.411742 −9 157 088.10 0.0201007 0.114832 0.297109 0.554298 −9 982 069.10 0.0506356 0.215795 0.484419 0.831212 −8 393 214.10 0.102587 0.351846 0.710723 1.145476 0.0157908 0.210721 0.584374 1.063623 1.61031 0.1015308 0.575364 1.212534 1.92256 2.67460 0.454936 1.38629 2.36597 3.35669 4.35146 10 0.675727 0.989256 1.34441 1.73493 2.15586 0.872090 1.239043 1.64650 2.08790 2.55821 1.23734 1.68987 2.17973 2.70039 3.24697 1.63538 2.16735 2.73264 3.32511 3.94030 2.20413 2.83311 3.48954 4.16816 4.86518 3.45460 4.25485 5.07064 5.89883 6.73720 5.34812 6.34581 7.34412 8.34283 9.34182 11 12 13 14 15 2.60322 3.07382 3.56503 4.07467 4.60092 3.05348 3.57057 4.10692 4.66043 5.22935 3.81575 4.40379 5.00875 5.62873 6.26214 4.57481 5.22603 5.89186 6.57063 7.26094 5.57778 6.30380 7.04150 7.78953 8.54676 7.58414 8.43842 9.29907 10.1653 11.0365 10.3410 11.3403 12.3398 13.3393 14.3389 16 17 18 19 20 5.14221 5.69722 6.26480 6.84397 7.43384 5.81221 6.40776 7.01491 7.63273 8.26040 6.90766 7.56419 8.23075 8.90652 9.59078 7.96165 8.67176 9.39046 10.1170 10.8508 9.31224 10.0852 10.8649 11.6509 12.4426 11.9122 12.7919 13.6753 14.5620 15.4518 15.3385 16.3382 17.3379 18.3377 19.3374 21 22 23 24 25 8.03365 8.64272 9.26043 9.88623 10.5197 8.89720 9.54249 10.19567 10.8564 11.5240 10.28293 10.9823 11.6886 12.4012 13.1197 11.5913 12.3380 13.0905 13.8484 14.6114 13.2396 14.0415 14.8480 15.6587 16.4734 16.3444 17.2396 18.1373 19.0373 19.9393 20.3372 21.3370 22.3369 23.3367 24.3266 26 27 28 29 30 11.1602 11.8076 12.4613 13.1211 13.7867 12.1981 12.8785 13.5647 14.2565 14.9535 13.8439 14.5734 15.3079 16.0471 16.7908 15.3792 16.1514 16.9279 17.7084 18.4927 17.2919 18.1139 18.9392 19.7677 20.5992 20.8434 21.7494 22.6572 23.5666 24.4776 25.3365 26.3363 27.3362 28.3361 29.3360 40 50 60 70 80 90 100 20.7065 27.9907 35.5345 43.2752 51.1719 59.1963 67.3276 22.1643 29.7067 37.4849 45.4417 53.5401 61.7541 70.0649 24.4330 32.3574 40.4817 48.7576 57.1532 65.6466 74.2219 26.5093 34.7643 43.1880 51.7393 60.3915 69.1260 77.9295 29.0505 37.6886 46.4589 55.3289 64.2778 73.2911 82.3581 33.6603 42.9421 52.2938 61.6983 71.1445 80.6247 90.1332 39.3353 49.3349 59.3347 69.3345 79.3343 89.3342 99.3341 416 0.990 0.975 0.950 STFE_Z02.qxd 26/02/2009 09:20 Page 417 Appendix: Tables v 0.250 0.100 0.050 0.025 0.010 0.005 0.001 1.32330 2.77259 4.10834 5.38527 6.62568 2.70554 4.60517 6.25139 7.77944 9.23636 003.84146 5.99146 7.81473 9.48773 11.0705 5.02389 7.37776 9.34840 11.1433 12.8325 6.63490 9.21034 11.3449 13.2767 15.0863 7.87944 10.5966 12.8382 14.8603 16.7496 10.828 13.816 16.266 18.467 20.515 10 7.84080 9.03715 10.2189 11.3888 12.5489 10.6446 12.0170 13.3616 14.6837 15.9872 12.5916 14.0671 15.5073 16.9190 18.3070 14.4494 16.0128 17.5345 19.0228 20.4832 16.8119 18.4753 20.0902 21.6660 23.2093 18.5476 20.2777 21.9550 23.5894 25.1882 22.458 24.322 26.125 27.877 29.588 11 12 13 14 15 13.7007 14.8454 15.9839 17.1169 18.2451 17.2750 18.5493 19.8119 21.0641 22.3071 19.6751 21.0261 22.3620 23.6848 24.9958 21.9200 23.3367 24.7356 26.1189 27.4884 24.7250 26.2170 27.6882 29.1412 30.5779 26.7568 28.2995 29.8195 31.3194 32.8013 31.264 32.909 34.528 36.123 37.697 16 17 18 19 20 19.3689 20.4887 21.6049 22.7178 23.8277 23.5418 24.7690 25.9894 27.2036 28.4120 26.2962 27.5871 28.8693 30.1435 31.4104 28.8454 30.1910 31.5264 32.8523 34.1696 31.9999 33.4087 34.8053 36.1909 37.5662 34.2672 35.7185 37.1565 38.5823 39.9968 29.252 40.790 42.312 43.820 45.315 21 22 23 24 25 24.9348 26.40393 27.1413 28.2412 29.3389 29.6151 30.8133 32.0069 33.1962 34.3816 32.6706 33.9244 35.1725 36.4150 37.6525 35.4789 36.7807 38.0756 39.3641 40.6465 38.9322 40.2894 41.6384 42.9798 44.3141 41.4011 42.7957 44.1813 45.5585 46.9279 46.797 48.268 49.728 51.179 52.618 26 27 28 29 30 30.4346 31.5284 32.6205 33.7109 34.7997 35.5632 36.7412 37.9150 39.0875 40.2560 38.8851 40.1133 41.3371 42.5570 43.7730 41.9232 43.1945 44.4608 45.7223 46.9792 45.6417 46.9629 48.2782 49.5879 50.8922 48.2899 49.6449 50.9934 52.3356 53.6720 54.052 55.476 56.892 58.301 59.703 40 50 60 70 80 90 100 45.6160 56.3336 66.9815 77.5767 88.1303 98.6499 109.141 51.8051 63.1671 74.3970 85.5270 96.5782 107.565 118.498 55.7585 67.5048 79.0819 90.5312 101.879 113.145 124.342 59.3417 71.4202 83.2977 95.0232 106.629 118.136 129.561 63.6907 76.1539 88.3794 100.425 112.329 124.116 135.807 66.7660 79.4900 91.9517 104.215 116.321 128.299 140.169 73.402 86.661 99.607 112.317 124.839 137.208 149.449 417 STFE_Z02.qxd 26/02/2009 09:20 Page 418 Appendix: Tables Table A5(a) Critical values of the F distribution (upper 5% points) The entries in the table give the critical values of F cutting off 5% in the right-hand tail of the distribution v1 gives the degrees of freedom in the numerator, v2 those in the denominator v1 161.45 18.513 10.128 7.7086 6.6079 199.50 19.000 9.5521 6.9443 5.7861 215.71 19.164 9.2766 6.5914 5.4095 224.58 19.247 9.1172 6.3882 5.1922 230.16 19.296 9.0135 6.2561 5.0503 230.99 19.330 8.9406 6.1631 4.9503 236.77 19.353 8.8867 6.0942 4.8759 238.88 19.371 8.8452 6.0410 4.8183 240.54 19.385 8.8123 5.9988 4.7725 10 5.9874 5.5914 5.3177 5.1174 4.9646 5.1433 4.7374 4.4590 4.2565 4.1028 4.7571 4.3468 4.0662 3.8625 3.7083 4.5337 4.1203 3.8379 3.6331 3.4780 4.3874 3.9715 3.6875 3.4817 3.3258 4.2839 3.8660 3.5806 3.3738 3.2172 4.2067 3.7870 3.5005 3.2927 3.1355 4.1468 3.7257 3.4381 3.2296 3.0717 4.0990 3.6767 3.3881 3.1789 3.0204 11 12 13 14 15 4.8443 4.7472 4.6672 4.6001 4.5431 3.9823 3.8853 3.8056 3.7389 3.6823 3.5874 3.4903 3.4105 3.3439 3.2874 3.3567 3.2592 3.1791 3.1122 3.0556 3.2039 3.1059 3.0254 2.9582 2.9013 3.0946 2.9961 2.9153 2.8477 2.7905 3.0123 2.9134 2.8321 2.7642 2.7066 2.9480 2.8486 2.7669 2.6987 2.6408 2.8962 2.7964 2.7144 2.6458 2.5876 16 17 18 19 20 4.4940 4.4513 4.4139 4.3807 4.3512 3.6337 3.5915 3.5546 3.5219 3.4928 3.2389 3.1968 3.1599 3.1274 2.0984 3.0069 2.9647 2.9277 2.8951 2.8661 2.8524 2.8100 2.7729 2.7401 2.7109 2.7413 2.6987 2.6613 2.6283 2.5990 2.6572 2.6143 2.5767 2.5435 2.5140 2.5911 2.5480 2.5102 2.4768 2.4471 2.5377 2.4943 2.4563 2.4227 2.3928 21 22 23 24 25 4.3248 4.3009 4.2793 4.2597 4.2417 3.4668 3.4434 3.4221 3.4028 3.3852 3.0725 3.0491 3.0280 3.0088 2.9912 2.8401 2.8167 2.7955 2.7763 2.7587 2.6848 2.6613 2.6400 2.6307 2.6030 2.5727 2.5491 2.5277 2.5082 2.4904 2.4876 2.4638 2.4422 2.4226 2.4047 2.4205 2.3965 2.3748 2.3551 2.3371 2.3660 2.3419 2.3201 2.3002 2.2821 26 27 28 29 30 4.2252 4.2100 4.1960 4.1830 4.1709 3.3690 3.3541 3.3404 3.3277 3.3158 2.9752 2.9604 2.9467 2.9340 2.9223 2.7426 2.7278 2.7141 2.7014 2.6896 2.5868 2.5719 2.5581 2.5454 2.5336 2.4741 2.4591 2.4453 2.4324 2.4205 2.3883 2.3732 2.3593 2.3463 2.3343 2.3205 2.3053 2.2913 2.2783 2.2662 2.2655 2.2501 2.2360 2.2229 2.2107 40 60 120 ∞ 4.0847 4.0012 3.9201 3.8415 3.2317 3.1504 3.0718 2.9957 2.8387 2.7581 2.6802 2.6049 2.6060 2.5252 2.4472 2.3719 2.4495 2.3683 2.2899 2.2141 2.3359 2.2541 2.1750 2.0986 2.2490 2.1665 2.0868 2.0096 2.1802 2.0970 2.0164 1.9384 2.1240 2.0401 1.9588 1.8799 v2 418 STFE_Z02.qxd 26/02/2009 09:20 Page 419 Appendix: Tables v1 10 12 15 20 24 30 40 60 120 ∞ 241.88 19.396 8.7855 5.9644 4.7351 243.91 19.413 8.7446 5.9117 4.6777 245.95 19.429 8.7029 5.8578 4.6188 248.01 19.446 8.6602 5.8025 4.5581 249.05 19.454 8.6385 5.7744 4.5272 250.10 19.462 8.6166 5.7459 4.4957 251.14 19.471 8.5944 5.7170 4.4638 252.20 19.479 8.5720 5.6877 4.4314 253.25 19.487 8.5494 5.6581 4.3985 254.31 19.496 8.5264 5.6281 4.3650 10 4.0600 3.6365 3.3472 3.1373 2.9782 3.9999 3.5747 3.2839 3.0729 2.9130 3.9381 3.5107 3.2184 3.0061 2.8450 3.8742 3.4445 3.1503 2.9365 2.7740 3.8415 3.4105 3.1152 2.9005 2.7372 3.8082 3.3758 3.0794 2.8637 2.6996 3.7743 3.3404 3.0428 2.8259 2.6609 3.7398 3.3043 3.0053 2.7872 2.6211 3.7047 3.2674 2.9669 2.7475 2.5801 3.6689 3.2298 2.9276 2.7067 2.5379 11 12 13 14 15 2.8536 2.7534 2.6710 2.6022 2.5437 2.7876 2.6866 2.6037 2.5342 2.4753 2.7186 2.6169 2.5331 2.4630 2.4034 2.6464 2.5436 2.4589 2.3879 2.3275 2.6090 2.5055 2.4202 2.3487 2.2878 2.5705 2.4663 2.3803 2.3082 2.2468 2.5309 2.4259 2.3392 2.2664 2.2043 2.4901 2.3842 2.2966 2.2229 2.1601 2.4480 2.3410 2.2524 2.1778 2.1141 2.4045 2.2962 2.2064 2.1307 2.0658 16 17 18 19 20 2.4935 2.4499 2.4117 2.3779 2.3479 2.4247 2.3807 2.3421 2.3080 2.2776 2.3522 2.3077 2.2686 2.2341 2.2033 2.2756 2.2304 2.1906 2.1555 2.1242 2.2354 2.1898 2.1497 2.1141 2.0825 2.1938 2.1477 2.1071 2.0712 2.0391 2.1507 2.1040 2.0629 2.0264 1.9938 2.1058 2.0584 2.0166 1.9795 1.9464 2.0589 2.0107 1.9681 1.9302 1.8963 2.0096 1.9604 1.9168 1.8780 1.8432 21 22 23 24 25 2.3210 2.2967 2.2747 2.2547 2.2365 2.2504 2.2258 2.2036 2.1834 2.1649 2.1757 2.1508 2.1282 2.1077 2.0889 2.0960 2.0707 2.0476 2.0267 2.0075 2.0540 2.0283 2.0050 1.9838 1.9643 2.0102 1.9842 1.9605 1.9390 1.9192 1.9645 1.9380 1.9139 1.8920 1.8718 1.9165 1.8894 1.8648 1.8424 1.8217 1.8657 1.8380 1.8128 1.7896 1.7684 1.8117 1.7831 1.7570 1.7330 1.7110 26 27 28 29 30 2.2197 2.2043 2.1900 2.1768 2.1646 2.1479 2.1323 2.1179 2.1045 2.0921 2.0716 2.0558 2.0411 2.0275 2.0148 1.9898 1.9736 1.9586 1.9446 1.9317 1.9464 1.9299 1.9147 1.9005 1.8874 1.9010 1.8842 1.8687 1.8543 1.8409 1.8533 1.8361 1.8203 1.8055 1.7918 1.8027 1.7851 1.7689 1.7537 1.7396 1.7488 1.7306 1.7138 1.6981 1.6835 1.6906 1.6717 1.6541 1.6376 1.6223 40 60 120 ∞ 2.0772 1.9926 1.9105 1.8307 2.0035 1.9174 1.8337 1.7522 1.9245 1.8364 1.7505 1.6664 1.8389 1.7480 1.6587 1.5705 1.7929 1.7001 1.6084 1.5173 1.7444 1.6491 1.5543 1.4591 1.6928 1.5943 1.4952 1.3940 1.6373 1.5343 1.4290 1.3180 1.5766 1.4673 1.3519 1.2214 1.5089 1.3893 1.2539 1.0000 v2 419 STFE_Z02.qxd 26/02/2009 09:20 Page 420 Appendix: Tables Table A5(b) Critical values of the F distribution (upper 2.5% points) The entries in the table give the critical values of F cutting off 2.5% in the right-hand tail of the distribution v1 gives the degrees of freedom in the numerator, v2 in the denominator v1 647.79 38.506 17.443 12.218 10.007 799.50 39.000 16.044 10.649 8.4336 864.16 39.165 15.439 9.9792 7.7636 899.58 39.248 15.101 9.6045 7.3879 921.85 39.298 14.885 9.3645 7.1464 937.11 39.331 14.735 9.1973 6.9777 948.22 39.355 14.624 9.0741 6.8531 956.66 39.373 14.540 8.9796 6.7572 963.28 39.387 14.473 8.9047 6.6811 v2 10 8.8131 8.0727 7.5709 7.2093 6.9367 7.2599 6.5415 6.0595 5.7147 5.4564 6.5988 5.8898 5.4160 5.0781 4.8256 6.2272 5.5226 5.0526 4.7181 4.4683 5.9876 5.2852 4.8173 4.4844 4.2361 5.8198 5.1186 4.6517 4.3197 4.0721 5.6955 4.9949 4.5286 4.1970 3.9498 5.5996 4.8993 4.4333 4.1020 3.8549 5.5234 4.8232 4.3572 4.0260 3.7790 11 12 13 14 15 6.7241 6.5538 6.4143 6.2979 6.1995 5.2559 5.0959 4.9653 4.8567 4.7650 4.6300 4.4742 4.3472 4.2417 4.1528 4.2751 4.1212 3.9959 3.8919 3.8043 4.0440 3.8911 3.7667 3.6634 3.5764 3.8807 3.7283 3.6043 3.5014 3.4147 3.7586 3.6065 3.4827 3.3799 3.2934 3.6638 3.5118 3.3880 3.2853 3.1987 3.5879 3.4358 3.3120 3.2093 3.1227 16 17 18 19 20 6.1151 6.0420 5.9781 5.9216 5.8715 4.6867 4.6189 4.5597 4.5075 4.4613 4.0768 4.0112 3.9539 3.9034 3.8587 3.7294 3.6648 3.6083 3.5587 3.5147 3.5021 3.4379 3.3820 3.3327 3.2891 3.3406 3.2767 3.2209 3.1718 3.1283 3.2194 3.1556 3.0999 3.0509 3.0074 3.1248 3.0610 3.0053 2.9563 2.9128 3.0488 2.9849 2.9219 2.8801 2.8365 21 22 23 24 25 5.8266 5.7863 5.7498 5.7166 5.6864 4.4199 4.3828 4.3492 4.3187 4.2909 3.8188 3.7829 3.7505 3.7211 3.6943 3.4754 3.4401 3.4083 3.3794 3.3530 3.2501 3.2151 3.1835 3.1548 3.1287 3.0895 3.0546 3.0232 2.9946 2.9685 2.9686 2.9338 2.9023 2.8738 2.8478 2.8740 2.8392 2.8077 2.7791 2.7531 2.7977 2.7628 2.7313 2.7027 2.6766 26 27 28 29 30 5.6586 5.6331 5.6096 5.5878 5.5675 4.2655 4.2421 4.2205 4.2006 4.1821 3.6697 3.6472 3.6264 3.6072 3.5894 3.3289 3.3067 3.2863 3.2674 3.2499 3.1048 3.0828 3.0626 3.0438 3.0265 2.9447 2.9228 2.9027 2.8840 2.8667 2.8240 2.8021 2.7820 2.7633 2.7460 2.7293 2.7074 2.6872 2.6686 2.6513 2.6528 2.6309 2.6106 2.5919 2.5746 40 60 120 ∞ 5.4239 5.2856 5.1523 5.0239 4.0510 3.9253 3.8046 3.6889 3.4633 3.3425 3.2269 3.1161 3.1261 3.0077 2.8943 2.7858 2.9037 2.7863 2.6740 2.5665 2.7444 2.6274 2.5154 2.4082 2.6238 2.5068 2.3948 2.2875 2.5289 2.4117 2.2994 2.1918 2.4519 2.3344 2.2217 2.1136 420 STFE_Z02.qxd 26/02/2009 09:20 Page 421 Appendix: Tables v1 30 40 60 120 ∞ 10 12 15 20 24 968.63 39.398 14.419 8.8439 6.6192 976.71 39.415 14.337 8.7512 6.5245 984.87 39.431 14.253 8.6565 6.4277 993.10 39.448 14.167 8.5599 6.3286 997.25 39.456 14.124 8.5109 6.2780 10 5.4613 4.7611 4.2951 3.9639 3.7168 5.3662 4.6658 4.1997 3.8682 3.6209 5.2687 4.5678 4.1012 3.7694 3.5217 5.1684 4.4667 3.9995 3.6669 3.4185 5.1172 4.4150 3.9472 3.6142 3.3654 5.0652 4.3624 3.8940 3.5604 3.3110 5.0125 4.3089 3.8398 3.5055 3.2554 4.9589 4.2544 3.7844 3.4493 3.1984 4.9044 4.1989 3.7279 3.3918 3.1399 4.8491 4.1423 3.6702 3.3329 3.0798 11 12 13 14 15 3.5257 3.3736 3.2497 3.1469 3.0602 3.4296 3.2773 3.1532 3.0502 2.9633 3.3299 3.1772 3.0527 2.9493 2.8621 3.2261 3.0728 2.9477 2.8437 2.7559 3.1725 3.0187 2.8932 2.7888 2.7006 3.1176 2.9633 2.8372 2.7324 2.6437 3.0613 2.9063 2.7797 2.6742 2.5850 3.0035 2.8478 2.7204 2.6142 2.5242 2.9441 2.7874 2.6590 2.5519 2.4611 2.8828 2.7249 2.5955 2.4872 2.3953 16 17 18 19 20 2.9862 2.9222 2.8664 2.8172 2.7737 2.8890 2.8249 2.7689 2.7196 2.6758 2.7875 2.7230 2.6667 2.6171 2.5731 2.6808 2.6158 2.5590 2.5089 2.4645 2.6252 2.5598 2.5027 2.4523 2.4076 2.5678 2.5020 2.4445 2.3937 2.3486 2.5085 2.4422 2.3842 2.3329 2.2873 2.4471 2.3801 2.3214 2.2696 2.2234 2.3831 2.3153 2.2558 2.2032 2.1562 2.3163 2.2474 2.1869 2.1333 2.0853 21 22 23 24 25 2.7348 2.6998 2.6682 2.6396 2.6135 2.6368 2.6017 2.5699 2.5411 2.5149 2.5338 2.4984 2.4665 2.4374 2.4110 2.4247 2.3890 2.3567 2.3273 2.3005 2.3675 2.3315 2.2989 2.2693 2.2422 2.3082 2.2718 2.2389 2.2090 2.1816 2.2465 2.2097 2.1763 2.1460 2.1183 2.1819 2.1446 2.1107 2.0799 2.0516 2.1141 2.0760 2.0415 2.0099 1.9811 2.0422 2.0032 1.9677 1.9353 1.9055 26 27 28 29 30 2.5896 2.5676 2.5473 2.5286 2.5112 2.4908 2.4688 2.4484 2.4295 2.4120 2.3867 2.3644 2.3438 2.3248 2.3072 2.2759 2.2533 2.2324 2.2131 2.1952 2.2174 2.1946 2.1735 2.1540 2.1359 2.1565 2.1334 2.1121 2.0923 2.0739 2.0928 2.0693 2.0477 2.0276 2.0089 2.0257 2.0018 1.9797 1.9591 1.9400 1.9545 1.9299 1.9072 1.8861 1.8664 1.8781 1.8527 1.8291 1.8072 1.7867 40 60 120 ∞ 2.3882 2.2702 2.1570 2.0483 2.2882 2.1692 2.0548 1.9447 2.1819 2.0613 1.9450 1.8326 2.0677 1.9445 1.8249 1.7085 2.0069 1.8817 1.7597 1.6402 1.9429 1.8152 1.6899 1.5660 1.8752 1.7440 1.6141 1.4835 1.8028 1.6668 1.5299 1.3883 1.7242 1.5810 1.4327 1.2684 1.6371 1.4821 1.3104 1.0000 v2 1001.4 1005.6 39.465 39.473 14.081 14.037 8.4613 8.4111 6.2269 6.1750 1009.8 39.481 13.992 8.3604 6.1225 1014.0 1018.3 39.400 39.498 13.947 13.902 8.3092 8.2573 6.069? 6.0153 421 STFE_Z02.qxd 26/02/2009 09:20 Page 422 Appendix: Tables Table A5(c) Critical values of the F distribution (upper 1% points) The entries in the table give the critical values of F cutting off 1% in the right-hand tail of the distribution v1 gives the degrees of freedom in the numerator, v2 in the denominator v1 5763.6 99.299 28.237 15.522 10.967 5859.0 99.333 27.911 15.207 10.672 5928.4 99.356 27.672 14.976 10.456 5981.1 99.374 27.489 14.799 10.289 6022.5 99.388 27.345 14.659 10.158 v2 4052.2 98.503 34.116 21.198 16.258 10 13.745 12.246 11.259 10.561 10.044 4999.5 99.000 30.817 18.000 13.274 5403.4 99.166 29.457 16.694 12.060 5624.6 99.249 28.710 15.977 11.392 10.925 9.5466 8.6491 8.0215 7.5594 9.7795 8.4513 7.5910 6.9919 6.5523 9.1483 7.8466 7.0061 6.4221 5.9943 8.7459 7.4604 6.6318 6.0569 5.6363 8.4661 7.1914 6.3707 5.8018 5.3858 8.2600 6.9928 6.1776 5.6129 5.2001 8.1017 6.8400 6.0289 5.4671 5.0567 7.9761 6.7188 5.9106 5.3511 4.9424 11 12 13 14 15 9.6460 9.3302 9.0738 8.8618 8.6831 7.2057 6.9266 6.7010 6.5149 6.3589 6.2167 5.9525 5.7394 5.5639 5.4170 5.6683 5.4120 5.2053 5.0354 4.8932 5.3160 5.0643 4.8616 4.6950 4.5556 5.0692 4.8206 4.6204 4.4558 4.3183 4.8861 4.6395 4.4410 4.2779 4.1415 4.7445 4.4994 4.3021 4.1399 4.0045 4.6315 4.3875 4.1911 4.0297 3.8948 16 17 18 19 20 8.5310 8.3997 8.2854 8.1849 8.0960 6.2262 6.1121 6.0129 5.9259 5.8489 5.2922 5.1850 5.0919 5.0103 4.9382 4.7726 4.6690 4.5790 4.5003 4.4307 4.4374 4.3359 4.2479 4.1708 4.1027 4.2016 4.1015 4.0146 3.9386 3.8714 4.0259 3.9267 3.8406 3.7653 3.6987 3.8896 3.7910 3.7054 3.6305 3.5644 3.7804 3.6822 3.5971 3.5225 3.4567 21 22 23 24 25 8.0166 7.9454 7.8811 7.8229 7.7698 5.7804 5.7190 5.6637 5.6136 5.5680 4.8740 4.8166 4.7649 4.7181 4.6755 4.3688 4.3134 4.2636 4.2184 4.1774 4.0421 3.9880 3.9392 3.8951 3.8550 3.8117 3.7583 3.7102 3.6667 3.6272 3.6396 3.5867 3.5390 3.4959 3.4568 3.5056 3.4530 3.4057 3.3629 3.3439 3.3981 3.3458 3.2986 3.2560 3.2172 26 27 28 29 30 7.7213 7.6767 7.6356 7.5977 7.5625 5.5263 5.4881 5.4529 5.4204 5.3903 4.6366 4.6009 4.5681 4.5378 4.5097 4.1400 4.1056 4.0740 4.0449 4.0179 3.8183 3.7848 3.7539 3.7254 3.6990 3.5911 3.5580 3.5276 3.4995 3.4735 3.4210 3.3882 3.3581 3.3303 3.3045 3.2884 3.2558 3.2259 3.1982 3.1726 3.1818 3.1494 3.1195 3.0920 3.0665 40 60 120 ∞ 7.3141 7.0771 6.8509 6.6349 5.1785 4.9774 4.7865 4.6052 4.3126 4.1259 3.9491 3.7816 3.8283 3.6490 3.4795 3.3192 3.5138 3.3389 3.1735 3.0173 3.2910 3.1187 2.9559 2.8020 3.1238 2.9530 2.7918 2.6393 2.9930 2.8233 2.6629 2.5113 2.8876 2.7185 2.5586 2.4073 422 STFE_Z02.qxd 26/02/2009 09:20 Page 423 Appendix: Tables v1 10 12 15 20 24 30 40 60 120 ∞ 6055.8 99.399 27.229 14.546 10.051 6106.3 99.416 27.052 14.374 9.8883 6157.3 99.433 26.872 14.198 9.7222 6208.7 99.449 26.690 14.020 9.5526 6234.6 99.458 26.598 13.929 9.4665 6260.6 99.466 26.505 13.838 9.3793 6286.8 99.474 26.411 13.745 9.2912 6313.0 99.482 26.316 13.652 9.2020 6339.4 99.491 26.221 13.558 9.1118 6365.9 99.499 26.125 13.463 9.0204 v2 10 7.8741 6.6201 5.8143 5.2565 4.8491 7.7183 6.4691 5.6667 5.1114 4.7059 7.5590 6.3143 5.5151 4.9621 4.5581 7.3958 6.1554 5.3591 4.8080 4.4054 7.3127 6.0743 5.2793 4.7290 4.3269 7.2285 5.9920 5.1981 4.6486 4.2469 7.1432 5.9084 5.1156 4.5666 4.1653 7.0567 5.8236 5.0316 4.4831 4.0819 6.9690 5.7373 4.9461 4.3978 3.9965 6.8800 5.6495 4.8588 4.3105 3.9090 11 12 13 14 15 4.5393 4.2961 4.1003 3.9394 3.8049 4.3974 4.1553 3.9603 3.8001 3.6662 4.2509 4.0096 3.8154 3.6557 3.5222 4.0990 3.8584 3.6646 3.5052 3.3719 4.0209 3.7805 3.5868 3.4274 3.2940 3.9411 3.7008 3.5070 3.3476 3.2141 3.8596 3.6192 3.4253 3.2656 3.1319 3.7761 3.5355 3.3413 3.1813 3.0471 3.6904 3.4494 3.2548 3.0942 2.9595 3.6024 3.3608 3.1654 3.0040 2.8684 16 17 18 19 20 3.6909 3.5931 3.5082 3.4338 3.3682 3.5527 3.4552 3.3706 3.2965 3.2311 3.4089 3.3117 3.2273 3.1533 3.0880 3.2587 3.1615 3.0771 3.0031 2.9377 3.1808 3.0835 2.9990 2.9249 2.8594 3.1007 2.0032 2.9185 2.8442 2.7785 3.0182 2.9205 2.8354 2.7608 2.6947 2.9330 2.8348 2.7493 2.6742 2.6077 2.8447 2.7459 2.6597 2.5839 2.5168 2.7528 2.6530 2.5660 2.4893 2.4212 21 22 23 24 25 3.3098 3.2576 3.2106 3.1681 3.1294 3.1730 3.1209 3.0740 3.0316 2.9931 3.0300 2.9779 2.9311 2.8887 2.8502 2.8796 2.8274 2.7805 2.7380 2.6993 2.8010 2.7488 2.7017 2.6591 2.6203 2.7200 2.6675 2.6202 2.5773 2.5383 2.6359 2.5831 2.5355 2.4923 2.4530 2.5484 2.4951 2.4471 2.4035 2.3637 2.4568 2.4029 2.3542 2.3100 2.2696 2.3603 2.3055 2.2558 2.2107 2.1694 26 27 28 29 30 3.0941 3.0618 3.0320 3.0045 2.9791 2.9578 2.9256 2.8959 2.8685 2.8431 2.8150 2.7827 2.7530 2.7256 2.7002 2.6640 2.6316 2.6017 2.5742 2.5487 2.5848 2.5522 2.5223 2.4946 2.4689 2.5026 2.4699 2.4397 2.4118 2.3860 2.4170 2.3840 2.3535 2.3253 2.2992 2.3273 2.2938 2.2629 2.2344 2.2079 2.2325 2.1985 2.1670 2.1379 2.1108 2.1315 2.0965 2.0642 2.0342 2.0062 40 60 120 ∞ 2.8005 2.6318 2.4721 2.3209 2.6648 2.4961 2.3363 2.1847 2.5216 2.3523 2.1915 2.0385 2.3689 2.1978 2.0346 1.8783 2.2880 2.1154 1.9500 1.7908 2.2034 2.0285 1.8600 1.6964 2.1142 1.9360 1.7628 1.5923 2.0194 1.8363 1.6557 1.4730 1.9172 1.7263 1.5330 1.3246 1.8047 1.6006 1.3805 1.0000 423 STFE_Z02.qxd 26/02/2009 09:20 Page 424 Appendix: Tables Table A5(d) Critical values of the F distribution (upper 0.5% points) The entries in the table give the critical values of F cutting off 0.5% in the right-hand tail of the distribution v1 gives the degrees of freedom in the numerator, v2 in the denominator v1 16 211 198.50 55.552 31.333 22.785 20 000 199.00 49.799 26.284 18.314 21 615 199.17 47.467 24.259 16.530 22 500 199.25 46.195 23.155 15.556 23 056 199.30 45.392 22.456 14.940 23 437 199.33 44.838 21.975 14.513 23 715 199.36 44.434 21.622 14.200 23 925 199.37 44.126 21.352 13.961 24 091 199.39 43.882 21.139 13.772 10 18.635 16.236 14.688 13.614 12.826 14.544 12.404 11.042 10.107 9.4270 12.917 10.882 9.5965 8.7171 8.0807 12.028 10.050 8.8051 7.9559 7.3428 11.464 9.5221 9.3018 7.4712 6.8724 11.073 9.1553 7.9520 7.1339 6.5446 10.786 8.8854 7.6941 6.8849 6.3025 10.566 8.6781 7.4959 6.6933 6.1159 10.391 8.5138 7.3386 6.5411 5.9676 11 12 13 14 15 12.226 11.754 11.374 11.060 10.798 8.9122 8.5096 8.1865 7.9216 7.7008 7.6004 7.2258 6.9258 6.6804 6.4760 6.8809 6.5211 6.2335 5.9984 5.8029 6.4217 6.0711 5.7910 5.5623 5.3721 6.1016 5.7570 5.4819 5.2574 5.0708 5.8648 5.5245 5.2529 5.0313 4.8473 5.6821 5.3451 5.0761 4.8566 4.6744 5.5368 5.2021 4.9351 4.7173 3.5364 16 17 18 19 20 10.575 10.384 10.218 10.073 9.9439 7.5138 7.3536 7.2148 7.0935 6.9865 6.3034 6.1556 6.0278 5.9161 5.8177 5.6378 5.4967 5.3746 5.2681 5.1743 5.2117 5.0746 3.9560 4.8526 4.7616 4.9134 4.7789 4.6627 4.5614 4.4721 4.6920 4.5594 4.4448 4.3448 4.2569 4.5207 4.3894 3.2759 4.1770 4.0900 4.3838 4.2535 4.1410 4.0428 3.9564 21 22 23 24 25 9.8295 9.7271 9.6348 9.5513 9.4753 6.8914 6.8064 6.7300 6.6609 6.5982 5.7304 5.6524 5.5823 5.5190 5.4615 5.0911 5.0168 4.9500 4.8898 4.8351 4.6809 4.6088 3.5441 4.4857 4.4327 4.3931 4.3225 4.2591 4.2019 4.1500 4.1789 4.1094 4.0469 3.9905 3.9394 4.0128 3.9440 3.8822 3.8264 3.7758 3.8799 3.8116 3.7502 3.6949 3.6447 26 27 28 29 30 9.4059 9.3423 9.2838 9.2297 9.1797 6.5409 6.4885 6.4403 6.3958 6.3547 5.4091 5.3611 5.3170 5.2764 5.2388 4.7852 4.7396 4.6977 4.6591 4.6234 4.3844 4.3402 4.2996 4.2622 4.2276 4.1027 4.0594 4.0197 3.9831 3.9492 3.8928 3.8501 3.8110 3.7749 3.7416 3.7297 3.6875 3.6487 3.6131 3.5801 3.5989 3.5571 3.5186 3.4832 3.4504 40 60 120 ∞ 8.8279 8.4946 8.1788 7.894 6.0664 5.7950 5.5393 5.2983 4.9758 4.7290 4.4972 4.2794 4.3738 4.1399 3.9207 3.7151 3.9860 3.7599 3.5482 3.3499 3.7129 3.4918 3.2849 3.0913 3.5088 3.2911 3.0874 2.8968 3.3498 3.1344 2.9330 2.7444 3.2220 3.0083 2.8083 2.6210 v2 424 STFE_Z02.qxd 26/02/2009 09:20 Page 425 Appendix: Tables v1 10 12 15 20 24 30 40 60 120 ∞ 24 224 199.40 43.686 20.967 13.618 24 426 199.42 43.387 20.705 13.384 24 630 199.43 43.085 20.438 13.146 24 836 199.45 42.778 20.167 12.903 24 940 199.46 42.622 20.030 12.780 25 044 199.47 42.466 19.892 12.656 25 148 199.47 42.308 19.752 12.530 25 253 199.48 42.149 19.611 12.402 25 359 199.49 41.989 19.468 12.274 25 464 199.50 41.828 19.325 12.144 v2 10 10.250 8.3803 7.2106 6.4172 5.8467 10.034 8.1764 7.0149 6.2274 5.6613 9.8140 7.9678 6.8143 6.0325 5.4707 9.5888 7.7540 6.6082 5.8318 5.2740 9.4742 7.6450 6.5029 5.7292 5.1732 9.3582 7.5345 6.3961 5.6248 5.0706 9.2408 7.4224 6.2875 5.5186 4.9659 9.1219 7.3088 6.1772 5.4104 4.8592 9.0015 7.1933 6.0649 5.3001 4.7501 8.8793 7.0760 5.9506 5.1875 4.6385 11 12 13 14 15 5.4183 5.0855 4.8199 4.6034 4.4235 5.2363 4.9062 4.6429 4.4281 4.2497 5.0489 4.7213 4.4600 4.2468 4.0698 4.8552 4.5299 4.2703 4.0585 3.8826 4.7557 4.4314 4.1726 3.9614 3.7859 4.6543 4.3309 4.0727 3.8619 3.6867 4.5508 4.2282 3.9704 3.7600 3.5850 4.4450 5.1229 3.8655 3.6552 3.4803 4.3367 4.0149 3.7577 3.5473 3.3722 4.2255 3.9039 3.6465 3.4359 3.2602 16 17 18 19 20 4.2719 4.1424 4.0305 3.9329 3.8470 4.0994 3.9709 3.8599 3.7631 3.6779 3.9205 3.7929 3.6827 4.5866 3.5020 3.7342 3.6073 3.4977 3.4020 3.3178 3.6378 3.5112 3.4017 3.3062 3.2220 3.5389 3.4124 3.3030 3.2075 3.1234 3.4372 3.3108 3.2014 3.1058 3.0215 3.3324 3.2058 3.0962 3.0004 2.9159 3.2240 3.0971 2.9871 2.8908 2.8058 3.1115 2.9839 2.8732 2.7762 2.6904 21 22 23 24 25 3.7709 3.7030 3.6420 3.5870 3.5370 3.6024 3.5350 3.4745 3.4199 3.3704 3.4270 3.3600 3.2999 3.2456 3.1963 3.2431 3.1764 3.1165 3.0624 3.0133 3.1474 3.0807 3.0208 2.9667 2.9176 3.0488 2.9821 2.9221 2.8679 2.8187 2.9467 2.8799 2.8197 2.7654 2.7160 2.7408 2.7736 2.7132 2.6585 2.6088 2.7302 2.6625 2.6015 2.5463 2.4961 2.6140 2.5455 2.4837 2.4276 2.3765 26 27 28 29 30 3.4916 3.4499 3.4117 3.3765 3.3440 3.3252 3.2839 3.2460 3.2110 3.1787 3.1515 3.1104 3.0727 3.0379 3.0057 2.9685 2.9275 2.8899 2.8551 2.8230 2.8728 2.8318 2.7941 2.7594 2.7272 2.7738 2.7327 2.6949 2.6600 2.6278 2.6709 2.6296 2.5916 2.5565 2.5241 2.5633 2.5217 2.4834 2.4479 2.4151 2.4501 2.4079 2.3690 2.3331 2.2998 2.3297 2.2867 2.2470 2.2102 2.1760 40 60 120 ∞ 3.1167 2.9042 2.7052 2.5188 2.9531 2.7419 2.5439 2.3583 2.7811 2.5705 2.3727 2.1868 2.5984 2.3872 2.1881 1.9998 2.5020 2.2898 2.0890 1.8983 2.4015 2.1874 1.9840 1.7891 2.2958 2.0789 1.8709 1.6691 2.1838 1.9622 1.7469 1.5325 2.0636 1.8341 1.6055 1.3637 1.9318 1.6885 1.4311 1.0000 425 STFE_Z02.qxd 26/02/2009 09:20 Page 426 Appendix: Tables Table A6 Critical values of Spearman’s rank correlation coefficient Entries in the table show critical values of Spearman’s rank correlation coefficient The value at the top of each column shows the significance level for a two-tail test For a one-tail test, the significance level is half that shown N 10% 5% 2% 1% 10 0.900 0.829 0.714 0.643 0.600 0.564 – 0.886 0.786 0.738 0.683 0.648 – 0.943 0.893 0.833 0.783 0.745 – – – 0.881 0.833 0.818 11 12 13 14 15 0.523 0.497 0.475 0.457 0.441 0.623 0.591 0.566 0.545 0.525 0.763 0.703 0.673 0.646 0.623 0.794 0.780 0.746 0.716 0.689 16 17 18 19 20 0.425 0.412 0.399 0.388 0.377 0.507 0.490 0.476 0.462 0.450 0.601 0.582 0.564 0.549 0.534 0.666 0.645 0.625 0.608 0.591 21 22 23 24 25 0.368 0.359 0.351 0.343 0.336 0.438 0.428 0.418 0.409 0.400 0.521 0.508 0.496 0.485 0.475 0.576 0.562 0.549 0.537 0.526 26 27 28 29 30 0.329 0.323 0.317 0.311 0.305 0.392 0.385 0.377 0.370 0.364 0.465 0.456 0.448 0.440 0.432 0.515 0.505 0.496 0.487 0.478 Source: Annals of Statistics, 1936 and 1949 426 STFE_Z02.qxd 26/02/2009 09:20 Page 427 Appendix: Tables Table A7 Critical values for the Durbin–Watson test at 5% significance level Sample size n 10 11 12 13 14 15 20 25 30 35 40 50 100 200 Number of explanatory variables dL 0.879 0.927 0.971 1.010 1.045 1.077 1.201 1.288 1.352 1.402 1.442 1.503 1.654 1.758 dU 1.320 1.324 1.331 1.340 1.350 1.361 1.411 1.454 1.489 1.519 1.544 1.585 1.694 1.778 dL 0.697 0.758 0.812 0.861 0.905 0.946 1.100 1.206 1.284 1.343 1.391 1.462 1.634 1.748 dU 1.641 1.604 1.579 1.562 1.551 1.543 1.537 1.550 1.567 1.584 1.600 1.628 1.715 1.789 dL 0.525 0.595 0.658 0.715 0.767 0.814 0.998 1.123 1.214 1.283 1.338 1.421 1.613 1.738 dU 2.016 1.928 1.864 1.816 1.779 1.750 1.676 1.654 1.650 1.653 1.659 1.674 1.736 1.799 dL 0.376 0.444 0.512 0.574 0.632 0.685 0.894 1.038 1.143 1.222 1.285 1.378 1.592 1.728 dU 2.414 2.283 2.177 2.094 2.030 1.977 1.828 1.767 1.739 1.726 1.721 1.721 1.758 1.810 dL 0.243 0.316 0.379 0.445 0.505 0.562 0.792 0.953 1.071 1.160 1.230 1.335 1.571 1.718 dU 2.822 2.645 2.506 2.390 2.296 2.220 1.991 1.886 1.833 1.803 1.786 1.771 1.780 1.820 427 ... Cataloging-in-Publication Data Barrow, Michael Statistics for economics, accounting and business studies / Michael Barrow – 5th ed p com Includes bibliographical references and index ISBN 978-0-273-71794-2... Statistics for Economics, Accounting and Business Studies The Power of Practice With your purchase of a new copy of this textbook, you received a Student Access Kit for getting started with statistics. .. between 16 and 19? 99 xiii STFE_A01.qxd 26/02/2009 09:01 Page xiv Getting started with statistics using MathXL This fifth edition of Statistics for Economics, Accounting and Business Studies comes