Modern Mathematical Statistics with Applications

Front Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Contents......Page 6
About the Authors......Page 4
Preface......Page 9
Introduction......Page 12
1.1 Populations and Samples......Page 13
Branches of Statistics......Page 15
Collecting Data......Page 18
1.2 Pictorial and Tabular Methods in Descriptive Statistics......Page 20
Stem-and-Leaf Displays......Page 21
Dotplots......Page 23
Histograms......Page 24
Histogram Shapes......Page 29
Qualitative Data......Page 30
Multivariate Data......Page 31
The Mean......Page 36
The Median......Page 38
Other Measures of Location: Quartiles, Percentiles, and Trimmed Means......Page 40
Categorical Data and Sample Proportions......Page 41
Measures of Variability for Sample Data......Page 44
Motivation for s[sup(2)]......Page 45
A Computing Formula for s[sup(2)]......Page 46
Boxplots......Page 48
Boxplots That Show Outliers......Page 50
Comparative Boxplots......Page 51
Supplementary Exercises......Page 56
Bibliography......Page 59
Introduction......Page 60
The Sample Space of an Experiment......Page 61
Events......Page 62
Some Relations from Set Theory......Page 63
2.2 Axioms, Interpretations, and Properties of Probability......Page 67
Interpreting Probability......Page 68
More Probability Properties......Page 70
Equally Likely Outcomes......Page 72
The Product Rule for Ordered Pairs......Page 76
Tree Diagrams......Page 77
Permutations......Page 78
Combinations......Page 80
2.4 Conditional Probability......Page 84
The Definition of Conditional Probability......Page 85
The Multiplication Rule for P(A B)......Page 87
Bayes’ Theorem......Page 89
2.5 Independence......Page 94
P(A B) When Events Are Independent......Page 95
Independence of More Than Two Events......Page 96
Supplementary Exercises......Page 100
Bibliography......Page 104
Introduction......Page 105
3.1 Random Variables......Page 106
Two Types of Random Variables......Page 108
3.2 Probability Distributions for Discrete Random Variables......Page 110
A Parameter of a Probability Distribution......Page 112
The Cumulative Distribution Function......Page 114
Another View of Probability Mass Functions......Page 117
3.3 Expected Values of Discrete Random Variables......Page 120
The Expected Value of X......Page 121
The Expected Value of a Function......Page 123
The Variance of X......Page 125
Rules of Variance......Page 126
3.4 Moments and Moment Generating Functions......Page 129
3.5 The Binomial Probability Distribution......Page 136
The Binomial Random Variable and Distribution......Page 138
Using Binomial Tables......Page 140
The Mean and Variance of X......Page 141
The Moment Generating Function of X......Page 142
The Hypergeometric Distribution......Page 145
The Negative Binomial Distribution......Page 149
3.7 The Poisson Probability Distribution......Page 153
The Poisson Distribution as a Limit......Page 154
The Mean, Variance and MGF of X......Page 155
The Poisson Process......Page 156
Supplementary Exercises......Page 160
Bibliography......Page 164
Introduction......Page 165
Probability Distributions for Continuous Variables......Page 166
The Cumulative Distribution Function......Page 170
Using F(x) to Compute Probabilities......Page 172
Obtaining f (x) from F(x)......Page 173
Percentiles of a Continuous Distribution......Page 174
Expected Values......Page 178
The Variance and Standard Deviation......Page 180
Moment Generating Functions......Page 181
4.3 The Normal Distribution......Page 186
The Standard Normal Distribution......Page 187
Percentiles of the Standard Normal Distribution......Page 189
z[sub(a)] Notation......Page 190
Nonstandard Normal Distributions......Page 191
The Normal Distribution and Discrete Populations......Page 194
Approximating the Binomial Distribution......Page 195
The Normal Moment Generating Function......Page 197
4.4 The Gamma Distribution and Its Relatives......Page 201
The Family of Gamma Distributions......Page 202
The Exponential Distribution......Page 204
The Chi-Squared Distribution......Page 207
The Weibull Distribution......Page 209
The Lognormal Distribution......Page 212
The Beta Distribution......Page 213
Sample Percentiles......Page 217
A Probability Plot......Page 218
Beyond Normality......Page 224
4.7 Transformations of a Random Variable......Page 227
Supplementary Exercises......Page 234
Bibliography......Page 239
Introduction......Page 240
The Joint Probability Mass Function for Two Discrete Random Variables......Page 241
The Joint Probability Density Function for Two Continuous Random Variables......Page 243
Independent Random Variables......Page 246
More Than Two Random Variables......Page 248
5.2 Expected Values, Covariance, and Correlation......Page 253
Covariance......Page 255
Correlation......Page 257
5.3 Conditional Distributions......Page 260
The Bivariate Normal Distribution......Page 265
Regression to the Mean......Page 267
The Mean and Variance via the Conditional Mean and Variance......Page 268
The Joint Distribution of Two New Random Variables......Page 273
The Joint Distribution of More Than Two New Variables......Page 276
5.5 Order Statistics......Page 278
The Distributions of Y[sub(n)] and Y[sub(1)]......Page 279
The Joint Distribution of the n Order Statistics......Page 281
The Distribution of a Single Order Statistic......Page 282
The Joint Distribution of Two Order Statistics......Page 283
An Intuitive Derivation of Order Statistic PDF’s......Page 284
Supplementary Exercises......Page 285
Bibliography......Page 288
Introduction......Page 289
6.1 Statistics and Their Distributions......Page 290
Random Samples......Page 292
Deriving the Sampling Distribution of a Statistic......Page 293
Simulation Experiments......Page 296
6.2 The Distribution of the Sample Mean......Page 302
The Case of a Normal Population Distribution......Page 303
The Central Limit Theorem......Page 304
Other Applications of the Central Limit Theorem......Page 307
The Law of Large Numbers......Page 308
6.3 The Distribution of a Linear Combination......Page 311
The Difference Between Two Random Variables......Page 313
The Case of Normal Random Variables......Page 314
Moment Generating Functions for Linear Combinations......Page 315
The Chi-Squared Distribution......Page 320
The t Distribution......Page 325
The F Distribution......Page 328
Summary of Relationships......Page 330
Supplementary Exercises......Page 332
Appendix: Proof of the Central Limit Theorem......Page 334
Introduction......Page 336
7.1 General Concepts and Criteria......Page 337
Mean Square Error......Page 339
Unbiased Estimators......Page 342
Estimators with Minimum Variance......Page 345
More Complications......Page 347
Reporting a Point Estimate: The Standard Error......Page 349
The Bootstrap......Page 350
The Method of Moments......Page 355
Maximum Likelihood Estimation......Page 357
Large-Sample Behavior of the MLE......Page 362
Some Complications......Page 363
7.3 Sufficiency......Page 366
The Factorization Theorem......Page 368
Jointly Sufficient Statistics......Page 370
Minimal Sufficiency......Page 371
Improving an Estimator......Page 372
Further Comments......Page 374
7.4 Information and Efficiency......Page 375
Information in a Random Sample......Page 377
The Cramér–Rao Inequality......Page 378
Large-Sample Properties of the MLE......Page 380
Supplementary Exercises......Page 383
Bibliography......Page 385
Introduction......Page 386
8.1 Basic Properties of Confidence Intervals......Page 387
Interpreting a Confidence Level......Page 389
Other Levels of Confidence......Page 390
Confidence Level, Precision, and Choice of Sample Size......Page 391
Deriving a Confidence Interval......Page 393
A Large-Sample Interval for m......Page 396
A Confidence Interval for a Population Proportion......Page 398
One-Sided Confidence Intervals (Confidence Bounds)......Page 401
Properties of t Distributions......Page 404
The One-Sample t Confidence Interval......Page 406
A Prediction Interval for a Single Future Value......Page 407
Tolerance Intervals......Page 409
Intervals Based on Nonnormal Population Distributions......Page 410
8.4 Confidence Intervals for the Variance and Standard Deviation of a Normal Population......Page 412
Bootstrapping the Mean......Page 415
Bootstrapping the Median......Page 419
The Mean Versus the Median......Page 420
Supplementary Exercises......Page 423
Bibliography......Page 427
Introduction......Page 428
9.1 Hypotheses and Test Procedures......Page 429
Test Procedures......Page 430
Errors in Hypothesis Testing......Page 431
Case I: A Normal Population with Known s......Page 439
Case II: Large-Sample Tests......Page 444
Case III: A Normal Population Distribution with Unknown s......Page 446
Large-Sample Tests......Page 453
Small-Sample Tests......Page 456
9.4 P-Values......Page 459
P-Values for z Tests......Page 462
P-Values for t Tests......Page 463
Statistical Versus Practical Significance......Page 467
Best Tests for Simple Hypotheses......Page 469
Power and Uniformly Most Powerful Tests......Page 472
Likelihood Ratio Tests......Page 474
Supplementary Exercises......Page 479
Bibliography......Page 482
Introduction......Page 483
10.1 z Tests and Confidence Intervals for a Difference Between Two Population Means......Page 484
Test Procedures for Normal Populations with Known Variances......Page 485
Using a Comparison to Identify Causality......Page 487
b and the Choice of Sample Size......Page 488
Large-Sample Tests......Page 489
Confidence Intervals for m1 m2......Page 492
10.2 The Two-Sample t Test and Confidence Interval......Page 498
Pooled t Procedures......Page 503
Type II Error Probabilities......Page 504
10.3 Analysis of Paired Data......Page 508
The Paired t Test......Page 510
A Confidence Interval for mD......Page 512
Paired Versus Unpaired Experiments......Page 514
10.4 Inferences About Two Population Proportions......Page 518
10.5 Inferences About Two Population Variances......Page 526
10.6 Comparisons Using the Bootstrap and Permutation Methods......Page 531
Introduction......Page 550
11.1 Single-Factor ANOVA......Page 551
Notation and Assumptions......Page 553
Sums of Squares and Mean Squares......Page 555
The F Test......Page 556
Computational Formulas......Page 557
Testing for the Assumption of Equal Variances......Page 560
Tukey’s Procedure......Page 563
The Interpretation of a in Tukey’s Procedure......Page 568
Confidence Intervals for Other Parametric Functions......Page 569
An Alternative Description of the ANOVA Model......Page 571
b for the F Test......Page 572
Relationship of the F Test to the t Test......Page 574
Single-Factor ANOVA When Sample Sizes Are Unequal......Page 575
Multiple Comparisons When Sample Sizes Are Unequal......Page 576
A Random Effects Model......Page 577
11.4 *Two-Factor ANOVA with Kij = 1......Page 581
The Model......Page 582
Test Procedures......Page 585
Expected Mean Squares......Page 587
Randomized Block Experiments......Page 588
Models for Random Effects......Page 591
Parameters for the Fixed Effects Model with Interaction......Page 595
Notation, Model, and Analysis......Page 596
Models with Mixed and Random Effects......Page 601
Supplementary Exercises......Page 606
Bibliography......Page 609
Introduction......Page 610
12.1 The Simple Linear and Logistic Regression Models......Page 611
A Linear Probabilistic Model......Page 614
The Logistic Regression Model......Page 618
12.2 Estimating Model Parameters......Page 622
Estimating s2 and s......Page 626
The Coefficient of Determination......Page 630
Terminology and Scope of Regression Analysis......Page 633
12.3 Inferences About the Regression Coefficient beta1......Page 637
A Confidence Interval for b1......Page 641
Hypothesis-Testing Procedures......Page 644
Regression and ANOVA......Page 646
Fitting the Logistic Regression Model......Page 647
12.4 Inferences Concerning mu gamma dot chi* and the Prediction of Future Upsilon Values......Page 651
Inferences Concerning mY#......Page 652
A Prediction Interval for a Future Value of Y......Page 655
The Sample Correlation Coefficient r......Page 659
Properties of r......Page 660
The Population Correlation Coefficient r and Inferences About Correlation......Page 662
Other Inferences Concerning r......Page 666
Residuals and Standardized Residuals......Page 671
Diagnostic Plots......Page 673
Difficulties and Remedies......Page 674
Estimating Parameters......Page 679
and the Coefficient of Multiple Determination s 2......Page 682
A Model Utility Test......Page 683
Inferences in Multiple Regression......Page 685
Assessing Model Adequacy......Page 687
Multiple Regression Models......Page 689
12.8 Regression with Matrices......Page 700
Covariance Matrices......Page 706
The Hat Matrix......Page 708
Supplementary Exercises......Page 713
Bibliography......Page 717
Introduction......Page 718
13.1 Goodness-of-Fit Tests When Category Probabilities Are Completely Specified......Page 719
x[sup(2)] When the pi’s Are Functions of Other Parameters......Page 723
x[sup(2)] When the Underlying Distribution Is Continuous......Page 724
13.2 Goodness-of-Fit Tests for Composite Hypotheses......Page 727
x[sup(2)] When Parameters Are Estimated......Page 728
Goodness of Fit for Discrete Distributions......Page 731
Goodness of Fit for Continuous Distributions......Page 734
A Special Test for Normality......Page 736
13.3 Two-Way Contingency Tables......Page 740
Testing for Homogeneity......Page 741
Testing for Independence......Page 743
Ordinal Factors and Logistic Regression......Page 745
Supplementary Exercises......Page 750
Bibliography......Page 753
Introduction......Page 754
14.1 The Wilcoxon Signed-Rank Test......Page 755
A General Description of the Wilcoxon Signed-Rank Test......Page 758
Paired Observations......Page 759
Efficiency of the Wilcoxon Signed-Rank Test......Page 760
Development of the Test When m=3, n=4......Page 763
General Description of the Rank-Sum Test......Page 764
Efficiency of the Wilcoxon Rank-Sum Test......Page 766
14.3 Distribution-Free Confidence Intervals......Page 768
The Wilcoxon Signed-Rank Interval......Page 769
The Wilcoxon Rank-Sum Interval......Page 771
14.4 Bayesian Methods......Page 773
14.5 Sequential Methods......Page 781
Sequential Testing for the Bernoulli Parameter......Page 782
The Expected Sample Size......Page 784
Supplementary Exercises......Page 788
Bibliography......Page 791
Appendix Tables......Page 792
A.1 Cumulative Binomial Probabilities......Page 793
A.2 Cumulative Poisson Probabilities......Page 796
A.3 Standard Normal Curve Areas......Page 797
A.4 The Incomplete Gamma Function......Page 799
A.5 Critical Values for t Distributions......Page 800
A.6 Tolerance Critical Values for Normal Population Distributions......Page 801
A.7 Critical Values for Chi-Squared Distributions......Page 802
A.8 t Curve Tail Areas......Page 803
A.9 Critical Values for F Distributions......Page 805
A.10 Critical Values for Studentized Range Distributions......Page 811
A.11 Chi-Squared Curve Tail Areas......Page 812
A.12 Critical Values for the Ryan–Joiner Test of Normality......Page 814
A.13 Critical Values for the Wilcoxon Signed-Rank Test......Page 815
A.14 Critical Values for the Wilcoxon Rank-Sum Test......Page 816
A.15 Critical Values for the Wilcoxon Signed-Rank Interval......Page 817
A.16 Critical Values for the Wilcoxon Rank-Sum Interval......Page 818
A.17 beta Curves for t Tests......Page 819
Answers to Odd-Numbered Exercises......Page 820
Index......Page 840