Devore J.L., Berk K.N. Modern Mathematical Statistics with Applications
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Springer Texts
in Statistics
Series Editors:
G. Casella
S. Fienberg
I. Olkin
For further volumes:
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Modern
Mathematical
Statistics with
Applications
Second Edition
Jay L. Devore
California Polytechnic State University
Kenneth N. Berk
Illinois State University
Jay L. Devore Kenneth N. Berk
California Polytechnic State University Illinois State University
Statistics Department Department of Mathematics
San Luis Obispo California Normal Illinois
USA USA
jdevore@calpoly.edu 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
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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 previ-
ously taught at the University of Florida and Oberlin College, and has had visiting
positions at Stanford, Harvard, the University of Washington, New York Univer-
sity, and Columbia. He has been at California Polytechnic State University,
San Luis Obi spo, since 1977, where he was chair of the Department of Statistics
for 7 years and recently achieved the exalted status of Professor Emeritus.
Jay has previously authored or coauthored five other books, including Probabil-
ity and Statistics for Engineering and the Sciences, which won a McGuffey
Longevity Award from the Text and Academic Authors Association for demon-
strated excellence over time. He is a Fellow of the American Statistical Associa-
tion, has been an associate editor for both the Journal of the American Statistical
Association and The American Statis tician, and received the Distinguished Teach-
ing 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 Statisti-
cian and edited it for 6 years. He served as secret ary/treasurer, program chair, and
chair of the Statistical Computing Section of the American Statistical Ass ociation,
and he twice co-chaired the Interface Sympo sium, 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
1 Overview and Descriptive Statistics 1
Introduction 1
1.1 Populations and Samples 2
1.2 Pictorial and Tabular Methods in Descriptive Statistics 9
1.3 Measures of Location 24
1.4 Measures of Variability 32
2 Probability 50
Introduction 50
2.1 Sample Spaces and Events 51
2.2 Axioms, Interpretations, and Properties of Probability 56
2.3 Counting Techniques 66
2.4 Conditional Probability 74
2.5 Independence 84
3 Discrete Random Variables and Probability Distributions 96
Introduction 96
3.1 Random Variables 97
3.2 Probability Distributions for Discrete Random Variables 101
3.3 Expected Values of Discrete Random Variables 112
3.4 Moments and Moment Generating Functions 121
3.5 The Binomial Probability Distribution 128
3.6 Hypergeometric and Negative Binomial Distributions 138
3.7 The Poisson Probability Distribution 146
4 Continuous Random Variables and Probability Distributions 158
Introduction 158
4.1 Probability Density Functions and Cumulative Distribution Functions 159
4.2 Expected Values and Moment Generating Functions 171
4.3 The Normal Distribution 179
4.4 The Gamma Distribution and Its Relatives 194
4.5 Other Continuous Distributions 202
4.6 Probability Plots 210
4.7 Transformations of a Random Variable 220
5 Joint Probability Distributions 232
Introduction 232
5.1 Jointly Distributed Random Variables 233
5.2 Expected Values, Covariance, and Correlation 245
5.3 Conditional Distributions 253
5.4 Transformations of Random Variables 265
5.5 Order Statistics 271
vii
6 Statistics and Sampling Distributions 284
Introduction 284
6.1 Statistics and Their Distributions 285
6.2 The Distribution of the Sample Mean 296
6.3 The Mean, Variance, and MGF for Several Variables 306
6.4 Distributions Based on a Normal Random Sample 315
Appendix: Proof of the Central Limit Theorem 329
7 Point Estimation 331
Introduction 331
7.1 General Concepts and Criteria 332
7.2 Methods of Point Estimation 350
7.3 Sufficiency 361
7.4 Information and Efficiency 371
8 Statistical Intervals Based on a Single Sample 382
Introduction 382
8.1 Basic Properties of Confidence Intervals 383
8.2 Large-Sample Confidence Intervals for a Population Mean and Proportion 391
8.3 Intervals Based on a Normal Population Distribution 401
8.4 Confidence Intervals for the Variance and Standard Deviation of a Normal
Population 409
8.5 Bootstrap Confidence Intervals 411
9 Tests of Hypotheses Based on a Single Sample 425
Introduction 425
9.1 Hypotheses and Test Procedures 426
9.2 Tests About a Population Mean 436
9.3 Tests Concerning a Population Proportion 450
9.4 P-Values 456
9.5 Some Comments on Selecting a Test Procedure 467
10 Inferences Based on Two Samples 484
Introduction 484
10.1 z Tests and Confidence Intervals for a Difference Between Two
Population Means 485
10.2 The Two-Sample t Test and Confidence Interval 499
10.3 Analysis of Paired Data 509
10.4 Inferences About Two Population Proportions 519
10.5 Inferences About Two Population Variances 527
10.6 Comparisons Using the Bootstrap and Permutation Methods 532
11 The Analysis of Variance 552
Introduction 552
11.1 Single-Factor ANOVA 553
11.2 Multiple Comparisons in ANOVA 564
11.3 More on Single-Factor ANOVA 572
11.4 Two-Factor ANOVA with K
ij
¼ 1 582
11.5 Two-Factor ANOVA with K
ij
> 1 597
12 Regression and Correlation 613
Introduction 613
12.1 The Simple Linear and Logistic Regression Models 614
12.2 Estimating Model Parameters 624
12.3 Inferences About the Regression Coefficient b
1
640
viii
Contents
12.4 Inferences Concerning m
Y x
and the Prediction of Future Y Values 654
12.5 Correlation 662
12.6 Assessing Model Adequacy 674
12.7 Multiple Regression Analysis 682
12.8 Regression with Matrices 705
13 Goodness-of-Fit Tests and Categorical Data Analysis 723
Introduction 723
13.1 Goodness-of-Fit Tests When Category Probabilities
Are Completely Specified 724
13.2 Goodness-of-Fit Tests for Composite Hypotheses 732
13.3 Two-Way Contingency Tables 744
14 Alternative Approaches to Inference 758
Introduction 758
14.1 The Wilcoxon Signed-Rank Test 759
14.2 The Wilcoxon Rank-Sum Test 766
14.3 Distribution-Free Confidence Intervals 771
14.4 Bayesian Methods 776
Appendix Tables 787
A.1 Cumulative Binomial Probabilities 788
A.2 Cumulative Poisson Probabilities 790
A.3 Standard Normal Curve Areas 792
A.4 The Incomplete Gamma Function 794
A.5 Critical Values for t Distributions 795
A.6 Critical Values for Chi-Squared Distributions 796
A.7 t Curve Tail Areas 797
A.8 Critical Values for F Distributions 799
A.9 Critical Values for Studentized Range Distributions 805
A.10 Chi-Squared Curve Tail Areas 806
A.11 Critical Values for the Ryan–Joiner Test of Normality 808
A.12 Critical Values for the Wilcoxon Signed-Rank Test 809
A.13 Critical Values for the Wilcoxon Rank-Sum Test 810
A.14 Critical Values for the Wilcoxon Signed-Rank Interval 811
A.15 Critical Values for the Wilcoxon Rank-Sum Interval 812
A.16 b Curves for t Tests 813
Answers to Odd-Numbered Exercises 814
Index 835
Contents ix