A I N NTRODUCTION TO M S ATHEMATICAL TATISTICS I A AND TS PPLICATIONS Fifth Edition Richard J. Larsen Vanderbilt University Morris L. Marx University of West Florida PrenticeHall Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo EditorinChief:DeirdreLynch AcquisitionsEditor:ChristopherCummings AssociateEditor:ChristinaLepre AssistantEditor:DanaJones SeniorManagingEditor:KarenWernholm AssociateManagingEditor:TamelaAmbush SeniorProductionProjectManager:PeggyMcMahon SeniorDesignSupervisor:AndreaNix CoverDesign:BethPaquin InteriorDesign:TamaraNewnam MarketingManager:AlexGay MarketingAssistant:KathleenDeChavez SeniorAuthorSupport/TechnologySpecialist:JoeVetere ManufacturingManager:EvelynBeaton SeniorManufacturingBuyer:CarolMelville ProductionCoordination,TechnicalIllustrations,andComposition:IntegraSoftwareServices,Inc. CoverPhoto:©JasonReed/GettyImages Manyofthedesignationsusedbymanufacturersandsellerstodistinguishtheirproductsareclaimedas trademarks.Wherethosedesignationsappearinthisbook,andPearsonwasawareofatrademark claim,thedesignationshavebeenprintedininitialcapsorallcaps. LibraryofCongressCataloging-in-PublicationData Larsen,RichardJ. Anintroductiontomathematicalstatisticsanditsapplications/ RichardJ.Larsen,MorrisL.Marx.—5thed. p.cm. Includesbibliographicalreferencesandindex. ISBN978-0-321-69394-5 1. Mathematicalstatistics—Textbooks. I.Marx,MorrisL. II. Title. QA276.L3142012 519.5—dc22 2010001387 Copyright©2012,2006,2001,1986,and1981byPearsonEducation,Inc.Allrightsreserved.Nopartof thispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyany means,electronic,mechanical,photocopying,recording,orotherwise,withoutthepriorwritten permissionofthepublisher.PrintedintheUnitedStatesofAmerica.Forinformationonobtaining permissionforuseofmaterialinthiswork,pleasesubmitawrittenrequesttoPearsonEducation,Inc., RightsandContractsDepartment,501BoylstonStreet,Suite900,Boston,MA02116,faxyourrequest to617-671-3447,ore-mailathttp://www.pearsoned.com/legal/permissions.htm. 12345678910—EB—1413121110 ISBN-13:978-0-321-69394-5 ISBN-10: 0-321-69394-9 Table of Contents Preface viii 1 Introduction 1 1.1 AnOverview 1 1.2 SomeExamples 2 1.3 ABriefHistory 7 1.4 AChapterSummary 14 2 Probability 16 2.1 Introduction 16 2.2 SampleSpacesandtheAlgebraofSets 18 2.3 TheProbabilityFunction 27 2.4 ConditionalProbability 32 2.5 Independence 53 2.6 Combinatorics 67 2.7 CombinatorialProbability 90 2.8 TakingaSecondLookatStatistics(MonteCarloTechniques) 99 3 Random Variables 102 3.1 Introduction 102 3.2 BinomialandHypergeometricProbabilities 103 3.3 DiscreteRandomVariables 118 3.4 ContinuousRandomVariables 129 3.5 ExpectedValues 139 3.6 TheVariance 155 3.7 JointDensities 162 3.8 TransformingandCombiningRandomVariables 176 3.9 FurtherPropertiesoftheMeanandVariance 183 3.10 OrderStatistics 193 3.11 ConditionalDensities 200 3.12 Moment-GeneratingFunctions 207 3.13 TakingaSecondLookatStatistics(InterpretingMeans) 216 Appendix3.A.1MinitabApplications 218 iii iv TableofContents 4 Special Distributions 221 4.1 Introduction 221 4.2 ThePoissonDistribution 222 4.3 TheNormalDistribution 239 4.4 TheGeometricDistribution 260 4.5 TheNegativeBinomialDistribution 262 4.6 TheGammaDistribution 270 4.7 TakingaSecondLookatStatistics(MonteCarlo Simulations) 274 Appendix4.A.1MinitabApplications 278 Appendix4.A.2AProofoftheCentralLimitTheorem 280 5 Estimation 281 5.1 Introduction 281 5.2 EstimatingParameters:TheMethodofMaximumLikelihoodand theMethodofMoments 284 5.3 IntervalEstimation 297 5.4 PropertiesofEstimators 312 5.5 Minimum-VarianceEstimators:TheCramér-RaoLower Bound 320 5.6 SufficientEstimators 323 5.7 Consistency 330 5.8 BayesianEstimation 333 5.9 TakingaSecondLookatStatistics(BeyondClassical Estimation) 345 Appendix5.A.1MinitabApplications 346 6 Hypothesis Testing 350 6.1 Introduction 350 6.2 TheDecisionRule 351 6.3 TestingBinomialData—H :p=p 361 0 o 6.4 TypeIandTypeIIErrors 366 6.5 ANotionofOptimality:TheGeneralizedLikelihoodRatio 379 6.6 TakingaSecondLookatStatistics(StatisticalSignificanceversus “Practical”Significance) 382 TableofContents v 7 Inferences Based on the Normal Distribution 385 7.1 Introduction 385 7.2 Comparing Y−µ and Y−µ 386 σ/√n S/√n 7.3 DerivingtheDistributionof Y−µ 388 S/√n 7.4 DrawingInferencesAboutµ 394 7.5 DrawingInferencesAboutσ2 410 7.6 TakingaSecondLookatStatistics(TypeIIError) 418 Appendix7.A.1MinitabApplications 421 Appendix7.A.2SomeDistributionResultsforYandS2 423 Appendix7.A.3AProofthattheOne-SampletTestisaGLRT 425 Appendix7.A.4AProofofTheorem7.5.2 427 8 Types of Data: A Brief Overview 430 8.1 Introduction 430 8.2 ClassifyingData 435 8.3 TakingaSecondLookatStatistics(SamplesAreNot “Valid”!) 455 9 Two-Sample Inferences 457 9.1 Introduction 457 9.2 TestingH0:µX=µY 458 9.3 TestingH :σ2 σ2—TheFTest 471 0 X = Y 9.4 BinomialData:TestingH0:pX =pY 476 9.5 ConfidenceIntervalsfortheTwo-SampleProblem 481 9.6 TakingaSecondLookatStatistics(ChoosingSamples) 487 Appendix9.A.1ADerivationoftheTwo-SampletTest(AProofof Theorem9.2.2) 488 Appendix9.A.2MinitabApplications 491 10 Goodness-of-Fit Tests 493 10.1 Introduction 493 10.2 TheMultinomialDistribution 494 10.3 Goodness-of-FitTests:AllParametersKnown 499 10.4 Goodness-of-FitTests:ParametersUnknown 509 10.5 ContingencyTables 519 vi TableofContents 10.6 TakingaSecondLookatStatistics(Outliers) 529 Appendix10.A.1MinitabApplications 531 11 Regression 532 11.1 Introduction 532 11.2 TheMethodofLeastSquares 533 11.3 TheLinearModel 555 11.4 CovarianceandCorrelation 575 11.5 TheBivariateNormalDistribution 582 11.6 TakingaSecondLookatStatistics(HowNottoInterpret theSampleCorrelationCoefficient) 589 Appendix11.A.1MinitabApplications 590 Appendix11.A.2AProofofTheorem11.3.3 592 12 The Analysis of Variance 595 12.1 Introduction 595 12.2 TheFTest 597 12.3 MultipleComparisons:Tukey’sMethod 608 12.4 TestingSubhypotheseswithContrasts 611 12.5 DataTransformations 617 12.6 TakingaSecondLookatStatistics(PuttingtheSubjectof StatisticsTogether—TheContributionsofRonaldA.Fisher) 619 Appendix12.A.1MinitabApplications 621 Appendix12.A.2AProofofTheorem12.2.2 624 SSTR/(k–1) Appendix12.A.3TheDistributionof SSE/(n–k) WhenH1 isTrue 624 13 Randomized Block Designs 629 13.1 Introduction 629 13.2 TheFTestforaRandomizedBlockDesign 630 13.3 ThePairedtTest 642 13.4 TakingaSecondLookatStatistics(Choosingbetweena Two-SampletTestandaPairedtTest) 649 Appendix13.A.1MinitabApplications 653 14 Nonparametric Statistics 655 14.1 Introduction 656 14.2 TheSignTest 657 TableofContents vii 14.3 WilcoxonTests 662 14.4 TheKruskal-WallisTest 677 14.5 TheFriedmanTest 682 14.6 TestingforRandomness 684 14.7 TakingaSecondLookatStatistics(ComparingParametric andNonparametricProcedures) 689 Appendix14.A.1MinitabApplications 693 Appendix: Statistical Tables 696 Answers to Selected Odd-Numbered Questions 723 Bibliography 745 Index 753 Preface Thefirsteditionofthistextwaspublishedin1981.Eachsubsequentrevisionsince then has undergone more than a few changes. Topics have been added, com- puter software and simulations introduced, and examples redone. What has not changed over the years is our pedagogical focus. As the title indicates, this book is an introduction to mathematical statistics and its applications. Those last three words arenot an afterthought. Wecontinue tobelieve that mathematical statistics is best learned and most effectively motivated when presented against a back- drop of real-world examples and all the issues that those examples necessarily raise. We recognize that college students today have more mathematics courses to choose from than ever before because of the new specialties and interdisciplinary areas that continue to emerge. For students wanting a broad educational experi- ence,anintroductiontoagiventopicmaybeallthattheirschedulescanreasonably accommodate.Ourresponsetothatrealityhasbeentoensurethateacheditionof this text provides a more comprehensive and more usable treatment of statistics thandiditspredecessors. Traditionally, the focus of mathematical statistics has been fairly narrow—the subject’sobjectivehasbeentoprovidethetheoreticalfoundationforallofthevar- ious procedures that are used for describing and analyzing data. What it has not spoken to at much length are the important questions of which procedure to use in a given situation, and why. But those are precisely the concerns that every user of statistics must inevitably confront. To that end, adding features that can create a path from the theory of statistics to its practice has become an increasingly high priority. New to This Edition • Beginning with the third edition, Chapter 8, titled “Data Models,” was added. Itdiscussedsomeofthebasicprinciplesofexperimentaldesign,aswellassome guidelinesforknowinghowtobeginastatisticalanalysis.Inthisfifthedition,the DataModels(“TypesofData:ABriefOverview”)chapterhasbeensubstantially rewrittentomakeitsmainpointsmoreaccessible. • Beginning with the fourth edition, the end of each chapter except the first fea- turedasectiontitled“TakingaSecondLookatStatistics.”Manyofthesesections describe the ways that statistical terminology is often misinterpreted in what we see, hear, and read in our modern media. Continuing in this vein of interpre- tation, we have added in this fifth edition comments called “About the Data.” These sections are scattered throughout the text and are intended to encourage the reader to think critically about a data set’s assumptions, interpretations, and implications. • Many examples and case studies have been updated, while some have been deletedandothersadded. • Section 3.8, “Transforming and Combining Random Variables,” has been rewritten. viii Preface ix • Section 3.9, “Further Properties oftheMean and Variance,” now includes adis- cussionofcovariancessothatsumsofrandomvariablescanbedealtwithinmore generality. • Chapter5,“Estimation,”nowhasanintroductiontobootstrapping. • Chapter7,“InferencesBasedontheNormalDistribution,”hasnewmaterialon thenoncentralt distributionanditsroleincalculatingTypeIIerrorprobabilities. • Chapter 9, “Two-Sample Inferences,” has a derivation of Welch’s approx- imation for testing the differences of two means in the case of unequal variances. Wehopethatthechangesinthiseditionwillnotundothebestfeaturesofthe firstfour.Whatmadethetaskofcreatingthefiftheditionanenjoyableexperience wasthenatureofthesubjectitselfandthewaythatitcanbebeautifullyelegantand down-to-earth practical, all at the same time. Ultimately, our goal is to share with the reader at least some small measure of the affection we feel for mathematical statisticsanditsapplications. Supplements Instructor’s Solutions Manual. This resource contains worked-out solutions to all text exercises and is available for download from the Pearson Education InstructorResourceCenter. Student Solutions Manual ISBN-10: 0-321-69402-3; ISBN-13: 978-0-321- 69402-7.Featuringcompletesolutionstoselectedexercises,thisisagreattool forstudentsastheystudyandworkthroughtheproblemmaterial. Acknowledgments We would like to thank the following reviewers for their detailed and valuable comments,criticisms,andsuggestions: Dr.AberaAbay,RowanUniversity KyleSiegrist,UniversityofAlabamainHuntsville DitlevMonrad,UniversityofIllinoisatUrbana-Champaign VidhuS.Prasad,UniversityofMassachusetts,Lowell Wen-QingXu,CaliforniaStateUniversity,LongBeach KatherineSt.Clair,ColbyCollege YiminXiao,MichiganStateUniversity NicolasChristou,UniversityofCalifornia,LosAngeles DamingXu,UniversityofOregon MariaRizzo,OhioUniversity DimitrisPolitis,UniversityofCaliforniaatSanDiego Finally, we convey our gratitude and appreciation to Pearson Arts & Sciences Associate Editor for Statistics Christina Lepre; Acquisitions Editor Christopher Cummings; and Senior Production Project Manager Peggy McMahon, as well as