Basic Properties of Confidence Intervals71的置信區(qū)間的基本性質

Chapter7.StatisticalIntervalsBasedonaSingleSampleWeiqiLuo(駱偉祺)SchoolofSoftwareSunYat-SenUniversityEmail：Office：#A3137.1.BasicPropertiesofConfidenceIntervals7.2.Larger-SampleConfidenceIntervalsforaPopulationMeanandProportion7.3IntervalsBasedonaNormalPopulationDistribution7.4ConfidenceIntervalsfortheVarianceandStandardDeviationofaNormalPopulation2Chapter7:StatisticalIntervalsBasedonASingleSampleIntroductionApointestimationprovidesnoinformationabouttheprecisionandreliabilityofestimation.Forexample,usingthestatisticXtocalculateapointestimateforthetrueaveragebreakingstrength(g)ofpapertowelsofacertainbrand,andsupposethatX=9322.7.Becauseofsamplevariability,itisvirtuallyneverthecasethatX=μ.Thepointestimatesaysnothingabouthowcloseitmightbetoμ.Analternativetoreportingasinglesensiblevaluefortheparameterbeingestimatedistocalculateandreportanentireintervalofplausiblevalues—anintervalestimateorconfidenceinterval(CI)Chapter7Introduction3ConsideringaSimpleCaseSupposethattheparameterofinterestisapopulationmeanμandthatThepopulationdistributionisnormal.Thevalueofthepopulationstandarddeviationσ

isknown

Normalityofthepopulationdistributionisoftenareasonableassumption.Ifthevalueofμ

isunknown,itisimplausiblethatthevalueofσ

wouldbeavailable.Inlatersections,wewilldevelopmethodsbasedonlessrestrictiveassumptions.7.1BasicPropertiesofConfidenceIntervals4Example7.1

Industrialengineerswhospecializeinergonomicsareconcernedwithdesigningworkspaceanddevicesoperatedbyworkerssoastoachievehighproductivityandcomfort.Asampleofn

=

31trainedtypistswasselected,andthepreferredkeyboardheightwasdeterminedforeachtypist.Theresultingsampleaveragepreferredheightwas

80.0cm.Assumingthatpreferredheightisnormallydistributedwithσ

=

2.0cm.PleaseobtainaCIforμ,thetrueaveragepreferredheightforthepopulationofallexperiencedtypists.ConsiderarandomsampleX1,X2,…Xnfromthenormaldistributionwithmeanvalueμandstandarddeviationσ.Thenaccordingtothepropositioninpp.245,thesamplemeanisnormallydistributionwithexpectedvalueμandstandarddeviation7.1BasicPropertiesofConfidenceIntervals5Example7.1(Cont’)

7.1BasicPropertiesofConfidenceIntervals6wehaveExample7.1(Cont’)7.1BasicPropertiesofConfidenceIntervals7TheCIof95%is:InterpretingaCI:Itcanbeparaphrasedas“theprobabilityis0.95thattherandomintervalincludesorcoversthetruevalueofμ.Truevalueofμ(Fixed)Intervalnumberwithdifferentsamplemeans(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)CI(Random)…Example7.2(Ex.7.1Cont’)

Thequantitiesneededforcomputationofthe95%CIforaveragepreferredheightareδ=2,n=31and.Theresultingintervalis7.1BasicPropertiesofConfidenceIntervals8Thatis,wecanbehighlyconfidentthat79.3<μ<80.7.Thisintervalisrelativelynarrow,indicatingthatμhasbeenratherpreciselyestimated.DefinitionIfafterobservingX1=x1,X2=x2,…Xn=xn,wecomputetheobservedsamplemean.Theresultingfixedintervaliscalleda

95%confidenceintervalforμ.ThisCIcanbeexpressedeitherasoras

7.1BasicPropertiesofConfidenceIntervals9isa95%CIforμwitha95%confidenceLowerLimitUpperLimitOtherLevelsofConfidence

A100(1-α)%confidenceintervalforthemeanμofanormalpopulationwhenthevalueofσisknownisgivenby

or,Forinstance,the99%CIis

7.1BasicPropertiesofConfidenceIntervals10WhyisSymmetry?Refertopp.291Ex.81-α0-zα/2+zα/2P(a<z<b)=1-αExample7.3

Let’scalculateaconfidenceintervalfortrueaverageholediameterusingaconfidencelevelof90%.

Thisrequiresthat100(1-α)=90,fromwhichα=0.1andzα/2=z0.05=1.645.Thedesiredintervalisthen

7.1BasicPropertiesofConfidenceIntervals11ConfidenceLevel,Precision,andChoiceofSampleSize7.1BasicPropertiesofConfidenceIntervals12Higherconfidencelevel(largerzα/2)AwiderintervalThenthewidth(Precision)oftheCILargerα

AwiderintervalSmallernAwiderintervalIndependentofthesamplemeanGivenadesiredconfidencelevel(α)andintervalwidth(w),thenwecandeterminethenecessarysamplesizen,byReliabilityPrecisionExample7.4

Extensivemonitoringofacomputertime-sharingsystemhassuggestedthatresponsetimetoaparticulareditingcommandisnormallydistributedwithstandarddeviation25millisec.Anewoperatingsystemhasbeeninstalled,andwewishtoestimatethetrueaverageresponsetimeμforthenewenvironment.Assumingthatresponsetimesarestillnormallydistributedwithσ=25,whatsamplesizeisnecessarytoensurethattheresulting95%CIhasawidthofnomorethan10?Thesamplesizenmustsatisfy

7.1BasicPropertiesofConfidenceIntervals13Sincenmustbeaninteger,asamplesizeof97isrequired.DerivingaConfidenceInterval

InthepreviousderivationoftheCIfortheunknownpopulationmeanθ=μofanormaldistributionwithknownstandarddeviationσ,wehaveconstructedthevariableTwopropertiesoftherandomvariabledependingfunctionallyontheparametertobeestimated(i.e.,μ)havingthestandardnormalprobabilitydistribution,whichdoesnotdependonμ.7.1BasicPropertiesofConfidenceIntervals14TheGeneralizedCaseLetX1,X2,…,XndenoteasampleonwhichtheCIforaparameterθistobebased.Supposearandomvariableh(X1,X2,…,Xn;θ)satisfyingthefollowingtwopropertiescanbefound:ThevariabledependsfunctionallyonbothX1,X2,…,Xn

andθ.Theprobabilitydistributionofthevariabledoesnotdependonθoronanyotherunknownparameters.7.1BasicPropertiesofConfidenceIntervals15Inordertodeterminea100(1-α)%CIofθ,weproceedasfollows:Becauseofthesecondproperty,aandbdonotdependonθ.Inthenormalexample,wehada=-Zα/2andb=Zα/2Supposewecanisolateθintheinequation:Ingeneral,theformofthehfunctionissuggestedbyexaminingthedistributionofanappropriateestimator.

7.1BasicPropertiesofConfidenceIntervals16Soa100(1-α)%CIisExample7.5

Atheoreticalmodelsuggestthatthetimetobreakdownofaninsulatingflu