An Introduction to Mathematical Statistics and Its Applications

753 Pages·2018·6.41 MB·English

by  Richard J. Larsen

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
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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