Springer Texts in Statistics Series Editors: G Casella S Fienberg I Olkin For further volumes: http://www.springer.com/series/417 Modern Mathematical Statistics with Applications Second Edition Jay L Devore California Polytechnic State University Kenneth N Berk Illinois State University Jay L Devore California Polytechnic State University Statistics Department San Luis Obispo California USA jdevore@calpoly.edu Kenneth N Berk Illinois State University Department of Mathematics Normal Illinois USA kberk@ilstu.edu ISBN 978-1-4614-0390-6 e-ISBN 978-1-4614-0391-3 DOI 10.1007/978-1-4614-0391-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011936004 # Springer Science+Business Media, LLC 2012 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To my wife Carol whose continuing support of my writing efforts over the years has made all the difference To my wife Laura who, as a successful author, is my mentor and role model About the Authors Jay L Devore Jay Devore received a B.S in Engineering Science from the University of California, Berkeley, and a Ph.D in Statistics from Stanford University He previously taught at the University of Florida and Oberlin College, and has had visiting positions at Stanford, Harvard, the University of Washington, New York University, and Columbia He has been at California Polytechnic State University, San Luis Obispo, since 1977, where he was chair of the Department of Statistics for years and recently achieved the exalted status of Professor Emeritus Jay has previously authored or coauthored five other books, including Probability and Statistics for Engineering and the Sciences, which won a McGuffey Longevity Award from the Text and Academic Authors Association for demonstrated excellence over time He is a Fellow of the American Statistical Association, has been an associate editor for both the Journal of the American Statistical Association and The American Statistician, and received the Distinguished Teaching Award from Cal Poly in 1991 His recreational interests include reading, playing tennis, traveling, and cooking and eating good food Kenneth N Berk Ken Berk has a B.S in Physics from Carnegie Tech (now Carnegie Mellon) and a Ph.D in Mathematics from the University of Minnesota He is Professor Emeritus of Mathematics at Illinois State University and a Fellow of the American Statistical Association He founded the Software Reviews section of The American Statistician and edited it for years He served as secretary/treasurer, program chair, and chair of the Statistical Computing Section of the American Statistical Association, and he twice co-chaired the Interface Symposium, the main annual meeting in statistical computing His published work includes papers on time series, statistical computing, regression analysis, and statistical graphics, as well as the book Data Analysis with Microsoft Excel (with Patrick Carey) vi Contents Preface x Overview and Descriptive Statistics 1.1 1.2 1.3 1.4 56 Introduction 96 Random Variables 97 Probability Distributions for Discrete Random Variables 101 Expected Values of Discrete Random Variables 112 Moments and Moment Generating Functions 121 The Binomial Probability Distribution 128 Hypergeometric and Negative Binomial Distributions 138 The Poisson Probability Distribution 146 Continuous Random Variables and Probability Distributions 158 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Introduction 50 Sample Spaces and Events 51 Axioms, Interpretations, and Properties of Probability Counting Techniques 66 Conditional Probability 74 Independence 84 Discrete Random Variables and Probability Distributions 96 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Probability 50 2.1 2.2 2.3 2.4 2.5 Introduction Populations and Samples Pictorial and Tabular Methods in Descriptive Statistics Measures of Location 24 Measures of Variability 32 Introduction 158 Probability Density Functions and Cumulative Distribution Functions Expected Values and Moment Generating Functions 171 The Normal Distribution 179 The Gamma Distribution and Its Relatives 194 Other Continuous Distributions 202 Probability Plots 210 Transformations of a Random Variable 220 159 Joint Probability Distributions 232 5.1 5.2 5.3 5.4 5.5 Introduction 232 Jointly Distributed Random Variables 233 Expected Values, Covariance, and Correlation Conditional Distributions 253 Transformations of Random Variables 265 Order Statistics 271 245 vii viii Contents Statistics and Sampling Distributions 284 6.1 6.2 6.3 6.4 Point Estimation 331 7.1 7.2 7.3 7.4 8.5 10.2 10.3 10.4 10.5 10.6 Introduction 484 z Tests and Confidence Intervals for a Difference Between Two Population Means 485 The Two-Sample t Test and Confidence Interval 499 Analysis of Paired Data 509 Inferences About Two Population Proportions 519 Inferences About Two Population Variances 527 Comparisons Using the Bootstrap and Permutation Methods 532 The Analysis of Variance 552 11.1 11.2 11.3 11.4 11.5 12 Introduction 425 Hypotheses and Test Procedures 426 Tests About a Population Mean 436 Tests Concerning a Population Proportion 450 P-Values 456 Some Comments on Selecting a Test Procedure 467 Inferences Based on Two Samples 484 10.1 11 Introduction 382 Basic Properties of Confidence Intervals 383 Large-Sample Confidence Intervals for a Population Mean and Proportion Intervals Based on a Normal Population Distribution 401 Confidence Intervals for the Variance and Standard Deviation of a Normal Population 409 Bootstrap Confidence Intervals 411 Tests of Hypotheses Based on a Single Sample 425 9.1 9.2 9.3 9.4 9.5 10 Introduction 331 General Concepts and Criteria 332 Methods of Point Estimation 350 Sufficiency 361 Information and Efficiency 371 Statistical Intervals Based on a Single Sample 382 8.1 8.2 8.3 8.4 Introduction 284 Statistics and Their Distributions 285 The Distribution of the Sample Mean 296 The Mean, Variance, and MGF for Several Variables 306 Distributions Based on a Normal Random Sample 315 Appendix: Proof of the Central Limit Theorem 329 Introduction 552 Single-Factor ANOVA 553 Multiple Comparisons in ANOVA 564 More on Single-Factor ANOVA 572 Two-Factor ANOVA with Kij ¼ 582 Two-Factor ANOVA with Kij > 597 Regression and Correlation 613 12.1 12.2 12.3 Introduction 613 The Simple Linear and Logistic Regression Models 614 Estimating Model Parameters 624 Inferences About the Regression Coefficient b1 640 391 Contents 12.4 12.5 12.6 12.7 12.8 13 654 Goodness-of-Fit Tests and Categorical Data Analysis 723 13.1 13.2 13.3 14 Inferences Concerning mY Áx à and the Prediction of Future Y Values Correlation 662 Assessing Model Adequacy 674 Multiple Regression Analysis 682 Regression with Matrices 705 Introduction 723 Goodness-of-Fit Tests When Category Probabilities Are Completely Specified 724 Goodness-of-Fit Tests for Composite Hypotheses 732 Two-Way Contingency Tables 744 Alternative Approaches to Inference 758 14.1 14.2 14.3 14.4 Introduction 758 The Wilcoxon Signed-Rank Test 759 The Wilcoxon Rank-Sum Test 766 Distribution-Free Confidence Intervals 771 Bayesian Methods 776 Appendix Tables 787 A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10 A.11 A.12 A.13 A.14 A.15 A.16 Cumulative Binomial Probabilities 788 Cumulative Poisson Probabilities 790 Standard Normal Curve Areas 792 The Incomplete Gamma Function 794 Critical Values for t Distributions 795 Critical Values for Chi-Squared Distributions 796 t Curve Tail Areas 797 Critical Values for F Distributions 799 Critical Values for Studentized Range Distributions 805 Chi-Squared Curve Tail Areas 806 Critical Values for the Ryan–Joiner Test of Normality 808 Critical Values for the Wilcoxon Signed-Rank Test 809 Critical Values for the Wilcoxon Rank-Sum Test 810 Critical Values for the Wilcoxon Signed-Rank Interval 811 Critical Values for the Wilcoxon Rank-Sum Interval 812 b Curves for t Tests 813 Answers to Odd-Numbered Exercises 814 Index 835 ix 798 Appendix Tables Table A.7 t Curve Tail Areas (cont.) Appendix Tables Table A.8 Critical Values for F Distributions 799 800 Appendix Tables Table A.8 Critical Values for F Distributions (cont.) Appendix Tables Table A.8 Critical Values for F Distributions (cont.) 801 802 Appendix Tables Table A.8 Critical Values for F Distributions (cont.) Appendix Tables Table A.8 Critical Values for F Distributions (cont.) 803 804 Appendix Tables Table A.8 Critical Values for F Distributions (cont.) Appendix Tables Table A.9 Critical Values for Studentized Range Distributions 805 806 Appendix Tables Table A.10 Chi-Squared Curve Tail Areas Appendix Tables Table A.10 Chi-Squared Curve Tail Areas (cont.) 807 808 Appendix Tables Table A.11 Critical Values for the Ryan–Joiner Test of Normality Appendix Tables Table A.12 Critical Values for the Wilcoxon Signed-Rank Test 809 810 Appendix Tables Table A.13 Critical Values for the Wilcoxon Rank-Sum Test Appendix Tables Table A.14 Critical Values for the Wilcoxon Signed-Rank Interval 811 812 Appendix Tables Table A.15 Critical Values for the Wilcoxon Rank-Sum Interval ... Texts in Statistics Series Editors: G Casella S Fienberg I Olkin For further volumes: http://www.springer.com/series/417 Modern Mathematical Statistics with Applications Second Edition Jay L Devore. .. CO (g/mile) 13.8 118 18.3 149 32.2 232 32.5 236 J.L Devore and K.N Berk, Modern Mathematical Statistics with Applications, Springer Texts in Statistics, DOI 10.1007/978-1-4614-0391-3_1, # Springer... audience and level Mathematical Level The challenge for students at this level should lie with mastery of statistical concepts as well as with mathematical wizardry Consequently, the mathematical prerequisites