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An Introduction to Mathematical Statistics and Its Applications PDF

753 Pages·2018·6.41 MB·English
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A I N NTRODUCTION TO M S ATHEMATICAL TATISTICS I A AND TS PPLICATIONS Sixth Edition Richard J. Larsen Vanderbilt University Morris L. Marx University of West Florida Director,PortfolioManagement:DeirdreLynch CoursewarePortfolioManager:PatrickBarbera CoursewarePortfolioAssistant:JustinBilling ContentProducer:KathleenA.Manley ManagingProducer:KarenWernholm MediaProducer:AudraWalsh ProductMarketingManager:YvonneVannatta ProductMarketingAssistant:JenniferMyers SeniorAuthorSupport/TechnologySpecialist:JoeVetere CoverandTextDesign,ProductionCoordination,Composition,andIllustrations: iEnergizerAptara®,Ltd. CoverImage:GlowGreenCircleArcs©Shutterstock/R.T.Wohlstadter Copyright©2018,2012,2006PearsonEducation,Inc.AllRightsReserved.Printedinthe UnitedStatesofAmerica.Thispublicationisprotectedbycopyright,andpermissionshould beobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrieval system,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying, recording,orotherwise.Forinformationregardingpermissions,requestformsandthe appropriatecontactswithinthePearsonEducationGlobalRights&Permissions department,pleasevisitwww.pearsoned.com/permissions/. PEARSON,ALWAYSLEARNING,andMyStatLabareexclusivetrademarksownedby PearsonEducation,Inc.oritsaffiliatesintheU.S.and/orothercountries. Unlessotherwiseindicatedherein,anythird-partytrademarksthatmayappearinthiswork arethepropertyoftheirrespectiveownersandanyreferencestothird-partytrademarks, logosorothertradedressarefordemonstrativeordescriptivepurposesonly.Such referencesarenotintendedtoimplyanysponsorship,endorsement,authorization,or promotionofPearson’sproductsbytheownersofsuchmarks,oranyrelationshipbetween theownerandPearsonEducation,Inc.oritsaffiliates,authors,licenseesordistributors. LibraryofCongressCataloging-in-PublicationData Names:Larsen,RichardJ. | Marx,MorrisL. Title:Anintroductiontomathematicalstatisticsanditsapplications/ RichardJ.Larsen,VanderbiltUniversity,MorrisL.Marx,Universityof WestFlorida. Description:Sixthedition. | Boston:Pearson,[2018] | Includes bibliographicalreferencesandindex. Identifiers:LCCN2016035117 | ISBN9780134114217 Subjects:LCSH:Mathematicalstatistics—Textbooks. Classification:LCCQA276.L3142018 | DDC519.5—dc23LCrecordavailableat https://lccn.loc.gov/2016035117 1 16 ISBN13:978-0-13-411421-7 ISBN10:0-13-411421-3 Contents Preface viii 1 Introduction 1 1.1 AnOverview 1 1.2 SomeExamples 2 1.3 ABriefHistory 6 1.4 AChapterSummary 14 2 Probability 15 2.1 Introduction 15 2.2 SampleSpacesandtheAlgebraofSets 17 2.3 TheProbabilityFunction 26 2.4 ConditionalProbability 31 2.5 Independence 50 2.6 Combinatorics 65 2.7 CombinatorialProbability 89 2.8 TakingaSecondLookatStatistics(MonteCarloTechniques) 99 3 Random Variables 102 3.1 Introduction 102 3.2 BinomialandHypergeometricProbabilities 103 3.3 DiscreteRandomVariables 116 3.4 ContinuousRandomVariables 127 3.5 ExpectedValues 137 3.6 TheVariance 153 3.7 JointDensities 160 3.8 TransformingandCombiningRandomVariables 174 3.9 FurtherPropertiesoftheMeanandVariance 182 3.10 OrderStatistics 192 3.11 ConditionalDensities 199 3.12 Moment-GeneratingFunctions 206 3.13 TakingaSecondLookatStatistics(InterpretingMeans) 215 iii iv Contents 4 Special Distributions 218 4.1 Introduction 218 4.2 ThePoissonDistribution 219 4.3 TheNormalDistribution 235 4.4 TheGeometricDistribution 257 4.5 TheNegativeBinomialDistribution 259 4.6 TheGammaDistribution 267 4.7 TakingaSecondLookatStatistics(MonteCarloSimulations) 271 Appendix4.A.1 PropertiesofFrequently-Usedpdfs 274 Appendix4.A.2 AProofoftheCentralLimitTheorem 276 5 Estimation 278 5.1 Introduction 278 5.2 EstimatingParameters:TheMethodofMaximumLikelihoodandtheMethod ofMoments 280 5.3 IntervalEstimation 293 5.4 PropertiesofEstimators 308 5.5 Minimum-VarianceEstimators:TheCram´er-RaoLowerBound 316 5.6 SufficientEstimators 319 5.7 Consistency 326 5.8 BayesianEstimation 329 5.9 TakingaSecondLookatStatistics(BeyondClassicalEstimation) 341 6 Hypothesis Testing 343 6.1 Introduction 343 6.2 TheDecisionRule 344 6.3 TestingBinomialData—H0:p=po 353 6.4 TypeIandTypeIIErrors 359 6.5 ANotionofOptimality:TheGeneralizedLikelihoodRatio 375 6.6 TakingaSecondLookatHypothesisTesting(StatisticalSignificanceversus “Practical”Significance) 378 7 Inferences Based on the Normal Distribution 380 7.1 Introduction 380 7.2 Comparing Y−√μ and Y−√μ 381 σ/ n S/ n 7.3 DerivingtheDistributionof Y−√μ 383 S/ n Contents v 7.4 DrawingInferencesAboutμ 389 7.5 DrawingInferencesAboutσ2 404 7.6 TakingaSecondLookatStatistics(TypeIIError) 412 Appendix7.A.1 SomeDistributionResultsforYandS2 414 Appendix7.A.2 AProofThattheOne-Samplet TestIsaGLRT 416 Appendix7.A.3 AProofofTheorem7.5.2 418 8 Types of Data: A Brief Overview 421 8.1 Introduction 421 8.2 ClassifyingData 427 8.3 TakingaSecondLookatStatistics(WhySamplesAreNot“Valid”!) 448 9 Two-Sample Inferences 450 9.1 Introduction 450 9.2 TestingH :μ =μ 451 0 X Y 9.3 TestingH :σ2 =σ2—TheFTest 463 0 X Y 9.4 BinomialData:TestingH :p =p 468 0 X Y 9.5 ConfidenceIntervalsfortheTwo-SampleProblem 473 9.6 TakingaSecondLookatStatistics(ChoosingSamples) 478 Appendix9.A.1 ADerivationoftheTwo-Samplet Test(AProofof Theorem9.2.2) 480 10 Goodness-of-Fit Tests 483 10.1 Introduction 483 10.2 TheMultinomialDistribution 484 10.3 Goodness-of-FitTests:AllParametersKnown 488 10.4 Goodness-of-FitTests:ParametersUnknown 498 10.5 ContingencyTables 507 10.6 TakingaSecondLookatStatistics(Outliers) 517 11 Regression 520 11.1 Introduction 520 11.2 TheMethodofLeastSquares 520 11.3 TheLinearModel 543 11.4 CovarianceandCorrelation 563 vi Contents 11.5 TheBivariateNormalDistribution 570 11.6 TakingaSecondLookatStatistics(HowNottoInterprettheSample CorrelationCoefficient) 576 Appendix11.A.1 AProofofTheorem11.3.3 577 12 The Analysis of Variance 580 12.1 Introduction 580 12.2 TheF Test 582 12.3 MultipleComparisons:Tukey’sMethod 592 12.4 TestingSubhypotheseswithContrasts 596 12.5 DataTransformations 604 12.6 TakingaSecondLookatStatistics(PuttingtheSubject ofStatisticsTogether—TheContributionsof RonaldA.Fisher) 606 Appendix12.A.1 AProofofTheorem12.2.2 608 Appendix12.A.2 TheDistributionof SSTR/(k−1) WhenH IsTrue 608 SSE/(n−k) 1 13 Randomized Block Designs 613 13.1 Introduction 613 13.2 TheFTestforaRandomizedBlockDesign 614 13.3 ThePairedtTest 628 13.4 TakingaSecondLookatStatistics(ChoosingbetweenaTwo-Samplet Test andaPairedt Test) 634 14 Nonparametric Statistics 638 14.1 Introduction 638 14.2 TheSignTest 639 14.3 WilcoxonTests 645 14.4 TheKruskal-WallisTest 658 14.5 TheFriedmanTest 662 14.6 TestingforRandomness 665 14.7 TakingaSecondLookatStatistics(ComparingParametricand NonparametricProcedures) 669 Contents vii 15 Factorial Data (Available Online) 15-1 15.1 Introduction 15-1 15.2 TheTwo-FactorFactorial 15-4 15.3 SumsofSquaresforTwo-FactorFactorials 15-16 15.4 ExpectedMeanSquares 15-26 15.5 Examples 15-30 15.6 TheThree-FactorFactorialDesign 15-40 15.7 2n Designs 15-51 15.8 FractionalFactorials 15-72 Appendix A: Statistical Tables 674 Answers to Selected Odd-Numbered Questions 701 Bibliography 725 Index 737 Preface JohnTukey(1915–2000)wasoneofthemostprominent(andquotable)statisticians ofthelasthalfofthetwentiethcentury.Hewasonceaskedwhatheespeciallyen- joyedabouttheprofessionhechoseforhislife’swork.“Thebestthingaboutbeing astatistician,”hesaidwithouthesitation,“isthatyougettoplayineveryone’sback- yard.”ThatsentimentsaysmuchaboutTukey’swellknowneclecticinterests;italso speakstowhatthisbookisallabout. Ourhopeisthatthistextsatisfiestwoobjectives:1)Itintroducesthebasictech- niquesofprobabilityandmathematicalstatisticsinacomprehensiveandinteresting wayatalevelappropriateforstudentswhohavecompletedthreesemestersofcal- culus,and2)itprovidesstudentswiththeskillsandinsightsnecessarytoapplythose principles.Inouropinion,satisfying(1)butnot(2)wouldbeaninadequate“take- away”forastudent’stwo-semestersworthoftimeandeffort. ItmayseemthatcompletingObjective1automaticallyconfersonstudentsthe wherewithaltomeetObjective2.Notso.Mathematicalstatisticsdealsprimarilywith thenatureofindividualmeasurementsorsimplepropertiescalculatedfromsamples of measurements—means, variances, distributions, relationships to other measure- ments,andsoon.Analyzingdata,though,requiresanadditionalknowledgeofthe experimentaldesigntowhichanentiresetofmeasurementsbelong.Toborrowsome terminologyfromeconomists,mathematicalstatisticsismuchlikethemicroaspect of the subject; experimental design is the macro aspect. There is enough time in a two-semestercoursetodojusticetoboth. Experimental designs come in many variations, but eight are especially impor- tantintermsofthefrequencyoftheiroccurrenceandtheirrelationshiptothemath- ematicalstatisticscoveredinafirstcourse.Theinitialstepinteachingsomeonehow to analyze data is helping them learn how to recognize those eight “data models”: One-sampledata,Two-sampledata,k-sampledata,Paireddata,Randomizedblock data,Regression/Correlationdata,Categoricaldata,andFactorialdata.Webelieve thatmentioningtheminpassingisnotsufficient.Theyneedtobecomparedandde- scribed,altogetherinonechapter,side-by-side,andillustratedwithreal-worlddata. Identifyingdatamodels,ofcourse,isnotadifficultskilltoacquire.Anyonein- volvedinanalyzingdatalearnsitquickly.Butforstudentstakingtheirfirstcoursein statistics,ignoringthetopicleavesthemwithoutasenseofwherethesubjectisgoing andwhy.FullyaddressingtheissuebeforestudentsencounteralltheZtests,t tests, χ2tests,andF teststhatcomeinrapidsuccessionprovidesaveryhelpfulframework forputtingallthatmaterialincontext. The final step in dealing with Objective 2 is to show the application of math- ematicalstatisticsandthemethodologiesitcreatedtoreal-worlddata.Made-upor contriveddatawillnotsuffice.Theydonotprovidethedetailorcomplexitynecessary topointout,forexample,whyoneparticulardesignwasusedratherthananotheror whattomakeofcertainanomaliesthatappeartohaveoccurredorwhatfollow-up studiesseemtobewarranted.Fortheirhelpandobviousexpertise,wearedeeplyin- debtedtoalltheresearcherswhohavegraciouslyallowedustouseportionsoftheir datatobasethemorethan80CaseStudiesscatteredthroughoutthetext.Wehope theseareasinformativeandhelpfulasthe“backyards”thatProfessorTukeyfound soenjoyable. viii Preface ix New to This Edition • Chapter15,FactorialData,isanew,downloadablechapterdescribingthethe- ory and practice of the analysis of variance as it applies to factorial data. It covers two-factor factorials, three-factor factorials, 2n designs, and fractional factorials, all at the same mathematical level as the book’s other two treat- mentsoftheanalysisofvariance,Chapter12andChapter13.Thisisthemost importantofallthemultifactorexperimentaldesigns. • Chapter2containstennewexamples,includingarepeated-independent-trials analysisoftheoften-quoted“Caesar’slastbreath”problem. • Overall,theSixthEditioncontains18newCaseStudiesforadditionalconcept application. • An Appendix has been added at the end of Chapter 4 summarizing all the importantpropertiesofthemostfrequentlyusedpdfs. • MuchofSection5.2dealingwithparameterestimationhasbeenrewritten,and themarginoferrorportionofSection5.3hasbeencompletelyredone. • DiscussionsofthedifferentdatamodelsinChapter8havebeenexpanded,and an eighth model (factorial data) has been added. The chapter includes seven newCaseStudies. • In Chapter 11, the section on nonlinear models has been thoroughly revised withanemphasisputontheirrelationshiptodifferentlawsofgrowth. • Becauseofspaceandcostconsiderations,journalsandtechnicalreportsoften display only summaries of an experiment’s results. A section has been added toChapter12showinghowtheentireANOVAtableforasetofk-sampledata canbe“reconstructed”withoutknowinganyoftheindividualmeasurements. • The text companion website www.pearsonhighered.com/mathstatsresources/ has the online, downloadable Chapter 15 and data sets analyzed in the text, ingenericformtocopyforinputintostatisticalsoftware.Thesitealsohasad- ditionalresourcestohelpstudentsandinstructors. Acknowledgments We would like to thank all of the following reviewers of the present and previous editionsofthistextfortheirmanyhelpfulcommentsandsuggestions.Everyedition benefitedfromyourinsightandyourexpertise. Reviewersforthisedition: AdamBowers,UniversityofCalifornia,SanDiego BreeEttinger,EmoryUniversity EugeneD.Gallagher,UniversityofMassachusetts,Boston MohammadKazemi,UniversityofNorthCarolina,Charlotte RalphRusso,UniversityofIowa NeslihanUler,UniversityofMichigan,AnnArbor BinWang,UniversityofSouthernAlabama Reviewersofpreviouseditions: AberaAbay,RowanUniversity KyleSiegrist,UniversityofAlabamainHuntsville DitlevMonrad,UniversityofIllinoisatUrbana-Champaign

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