Modern Mathematical Statistics with Applications

858 Pages·2011·5.95 MB·English

Springer Texts
in Statistics
Series Editors:
G. Casella
S. Fienberg
I. Olkin
Forfurther volumes:
http://www.springer.com/series/417

Modern
Mathematical
Statistics with
Applications
Second Edition
Jay L. Devore
CaliforniaPolytechnicStateUniversity
Kenneth N. Berk
IllinoisStateUniversity
JayL.Devore KennethN.Berk
CaliforniaPolytechnicStateUniversity IllinoisStateUniversity
StatisticsDepartment DepartmentofMathematics
SanLuisObispoCalifornia NormalIllinois
USA USA
[email protected] [email protected]
ISBN978-1-4614-0390-6 e-ISBN978-1-4614-0391-3
DOI10.1007/978-1-4614-0391-3
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2011936004
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To my wife Carol
whosecontinuingsupportofmywritingefforts
overtheyearshasmadeallthedifference.
To my wife Laura
who,asasuccessfulauthor,ismymentorandrolemodel.
About the Authors
Jay L. Devore
Jay Devore received a B.S. in Engineering Science from the University of
California,Berkeley,andaPh.D.inStatisticsfromStanfordUniversity.Heprevi-
ouslytaughtattheUniversityofFloridaandOberlinCollege,andhashadvisiting
positions at Stanford, Harvard, the University of Washington, New York Univer-
sity, and Columbia. He has been at California Polytechnic State University,
San Luis Obispo, since 1977, where he was chair of the Department of Statistics
for7yearsandrecentlyachievedtheexaltedstatusofProfessorEmeritus.
Jayhaspreviouslyauthoredorcoauthoredfiveotherbooks,includingProbabil-
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
AssociationandTheAmericanStatistician,andreceivedtheDistinguishedTeach-
ing Award from Cal Poly in 1991. His recreational interests include reading,
playingtennis,traveling,andcookingandeatinggoodfood.
Kenneth N. Berk
KenBerkhasaB.S.inPhysicsfromCarnegieTech(nowCarnegieMellon)anda
Ph.D.inMathematicsfromtheUniversityofMinnesota.HeisProfessorEmeritus
ofMathematicsatIllinoisStateUniversityandaFellowoftheAmericanStatistical
Association. He founded the Software Reviews section of The American Statisti-
cianandediteditfor6years.Heservedassecretary/treasurer,programchair,and
chairoftheStatisticalComputingSectionoftheAmericanStatisticalAssociation,
and he twice co-chaired the Interface Symposium, the main annual meeting in
statisticalcomputing.Hispublishedworkincludespapersontimeseries,statistical
computing, regression analysis, and statistical graphics, as well as the book Data
AnalysiswithMicrosoftExcel(withPatrickCarey).
vi
Contents
Preface x
1 Overview and Descriptive Statistics 1
Introduction 1
1.1 PopulationsandSamples 2
1.2 PictorialandTabularMethodsinDescriptiveStatistics 9
1.3 MeasuresofLocation 24
1.4 MeasuresofVariability 32
2 Probability 50
Introduction 50
2.1 SampleSpacesandEvents 51
2.2 Axioms,Interpretations,andPropertiesofProbability 56
2.3 CountingTechniques 66
2.4 ConditionalProbability 74
2.5 Independence 84
3 Discrete Random Variables and Probability Distributions 96
Introduction 96
3.1 RandomVariables 97
3.2 ProbabilityDistributionsforDiscreteRandomVariables 101
3.3 ExpectedValuesofDiscreteRandomVariables 112
3.4 MomentsandMomentGeneratingFunctions 121
3.5 TheBinomialProbabilityDistribution 128
3.6 HypergeometricandNegativeBinomialDistributions 138
3.7 ThePoissonProbabilityDistribution 146
4 Continuous Random Variables and Probability Distributions 158
Introduction 158
4.1 ProbabilityDensityFunctionsandCumulativeDistributionFunctions 159
4.2 ExpectedValuesandMomentGeneratingFunctions 171
4.3 TheNormalDistribution 179
4.4 TheGammaDistributionandItsRelatives 194
4.5 OtherContinuousDistributions 202
4.6 ProbabilityPlots 210
4.7 TransformationsofaRandomVariable 220
5 Joint Probability Distributions 232
Introduction 232
5.1 JointlyDistributedRandomVariables 233
5.2 ExpectedValues,Covariance,andCorrelation 245
5.3 ConditionalDistributions 253
5.4 TransformationsofRandomVariables 265
5.5 OrderStatistics 271
vii
viii Contents
6 Statistics and Sampling Distributions 284
Introduction 284
6.1 StatisticsandTheirDistributions 285
6.2 TheDistributionoftheSampleMean 296
6.3 TheMean,Variance,andMGFforSeveralVariables 306
6.4 DistributionsBasedonaNormalRandomSample 315
Appendix: ProofoftheCentralLimitTheorem 329
7 Point Estimation 331
Introduction 331
7.1 GeneralConceptsandCriteria 332
7.2 MethodsofPointEstimation 350
7.3 Sufficiency 361
7.4 InformationandEfficiency 371
8 Statistical Intervals Based on a Single Sample 382
Introduction 382
8.1 BasicPropertiesofConfidenceIntervals 383
8.2 Large-SampleConfidenceIntervalsforaPopulationMeanandProportion 391
8.3 IntervalsBasedonaNormalPopulationDistribution 401
8.4 ConfidenceIntervalsfortheVarianceandStandardDeviationofaNormal
Population 409
8.5 BootstrapConfidenceIntervals 411
9 Tests of Hypotheses Based on a Single Sample 425
Introduction 425
9.1 HypothesesandTestProcedures 426
9.2 TestsAboutaPopulationMean 436
9.3 TestsConcerningaPopulationProportion 450
9.4 P-Values 456
9.5 SomeCommentsonSelectingaTestProcedure 467
10 Inferences Based on Two Samples 484
Introduction 484
10.1 zTestsandConfidenceIntervalsforaDifferenceBetweenTwo
PopulationMeans 485
10.2 TheTwo-SampletTestandConfidenceInterval 499
10.3 AnalysisofPairedData 509
10.4 InferencesAboutTwoPopulationProportions 519
10.5 InferencesAboutTwoPopulationVariances 527
10.6 ComparisonsUsingtheBootstrapandPermutationMethods 532
11 The Analysis of Variance 552
Introduction 552
11.1 Single-FactorANOVA 553
11.2 MultipleComparisonsinANOVA 564
11.3 MoreonSingle-FactorANOVA 572
11.4 Two-FactorANOVAwithKij ¼ 1 582
11.5 Two-FactorANOVAwithKij > 1 597
12 Regression and Correlation 613
Introduction 613
12.1 TheSimpleLinearandLogisticRegressionModels 614
12.2 EstimatingModelParameters 624
12.3 InferencesAbouttheRegressionCoefficientb 640
1
Contents ix
12.4 InferencesConcerningmY(cid:2)x(cid:3) andthePredictionofFutureY Values 654
12.5 Correlation 662
12.6 AssessingModelAdequacy 674
12.7 MultipleRegressionAnalysis 682
12.8 RegressionwithMatrices 705
13 Goodness-of-Fit Tests and Categorical Data Analysis 723
Introduction 723
13.1 Goodness-of-FitTestsWhenCategoryProbabilities
AreCompletelySpecified 724
13.2 Goodness-of-FitTestsforCompositeHypotheses 732
13.3 Two-WayContingencyTables 744
14 Alternative Approaches to Inference 758
Introduction 758
14.1 TheWilcoxonSigned-RankTest 759
14.2 TheWilcoxonRank-SumTest 766
14.3 Distribution-FreeConfidenceIntervals 771
14.4 BayesianMethods 776
Appendix Tables 787
A.1 CumulativeBinomialProbabilities 788
A.2 CumulativePoissonProbabilities 790
A.3 StandardNormalCurveAreas 792
A.4 TheIncompleteGammaFunction 794
A.5 CriticalValuesfortDistributions 795
A.6 CriticalValuesforChi-SquaredDistributions 796
A.7 tCurveTailAreas 797
A.8 CriticalValuesforFDistributions 799
A.9 CriticalValuesforStudentizedRangeDistributions 805
A.10 Chi-SquaredCurveTailAreas 806
A.11 CriticalValuesfortheRyan–JoinerTestofNormality 808
A.12 CriticalValuesfortheWilcoxonSigned-RankTest 809
A.13 CriticalValuesfortheWilcoxonRank-SumTest 810
A.14 CriticalValuesfortheWilcoxonSigned-RankInterval 811
A.15 CriticalValuesfortheWilcoxonRank-SumInterval 812
A.16 bCurvesfortTests 813
Answers to Odd-Numbered Exercises 814
Index 835