An Introduction to Mathematical Statistics and Its Applications (2-downloads)

768 Pages·2014·15.34 MB·English

by  Richard Larsen

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
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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
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ISBN-13:978-0-321-69394-5
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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