CHAPTER-6

SamplingerrorandconfidenceintervalsCHAPTER-6

SamplingerrorandcpopulationsamplestatisticParametererrorpopulationsamplestatisticParamSection1samplingerrorofmeanSection2tdistributionSection3confidenceintervalsforthepopulationmeanSection1samplingerrorofSection1

samplingerrorofmean

Section1

samplingerroroAsimplerandomsampleisasampleofsizendrawnfromapopulationofsizeNinsuchawaythateverypossiblerandomsamplesnhasthesameprobabilityofbeingselected.Variabilityamongthesimplerandomsamplesdrawnfromthesamepopulationiscalledsamplingvariability,andtheprobabilitydistributionthatcharacterizessomeaspectofthesamplingvariability,usuallythemeanbutnotalways,iscalledasamplingdistribution.Thesesamplingdistributionsallowustomakeobjectivestatementsaboutpopulationparameterswithoutmeasuringeveryobjectinthepopulation.Asimplerandomsampleisa[Example1]ThepopulationmeanofDBPintheChineseadultmenis72mmHgwithstandarddeviation5mmHg.10adultparticipantswaschosenrandomlyfromtheChineseadultmen,herewecancalculatethesamplemeanandsamplestandarddeviation.Supposingsampling100times,what’stheresult?[Example1]linkageNlinkageNIfrandomsamplesarerepeatedlydrawnfromapopulationwithameanμandstandarddeviationσ,wecanfind:1thesamplemeansaredifferentfromtheothers2Thesamplemeanarenotnecessaryequaltopopulationmeanμ3ThedistributionofsamplemeanissymmetricaboutμHOWTOEXPLORETHESAMPLINGDISTRIBUTIONFORTHEMEAN?IfrandomsamplesarerepeaThedifferencebetweensamplestatisticsandpopulationparameterorthedifferenceamongsamplestatisticsarecalledsamplingerror.ThedifferencebetweensamplInreallifewesampleonlyonce,butwerealizethatoursamplecomesfromatheoreticalsamplingdistributionofallpossiblesamplesofaparticularsize.Thesamplingdistributionconceptprovidesalinkbetweensamplingvariabilityandprobability.Choosingarandomsampleisachanceoperationandgeneratingthesamplingdistributionconsistsofmanyrepetitionsofthischanceoperation.InreallifewesampleonlyonWhensamplingfromanormallydistributedpopulationwithmeanμ,thedistributionofthesamplemeanwillbenormalwithmeanμCentrallimitTheoremWhensamplingfromanormally

=50

=10XPopulationdistributionn=4SamplingdistributionXn=16=50=10XPopulationdistriWhensamplingfromanonnormallydistributedpopulationwithmeanμ,thedistributionofthesamplemeanwillbeapproximatelynormalwithmeanμaslongasnislargerenough(n>50).CentrallimitTheoremWhensamplingfromanonnormalXXStandarderror(SE)canbeusedtoassesssamplingerrorofmean.Althoughsamplingerrorisinevitable,itcanbecalculatedaccurately.Standarderror(SE)canbetheoreticalvalueofSEestimationofSECalculationofstandarderror(SE)s↑→SE↑n↑→SE↓linkagetheoreticalvalueofSEestimatExample5.2Oneanalystchoserandomlyasample(n=100)andmeasuredtheirweightswithameanof72kgandstandarddeviationof15kg.Question:whatisthestandarderror?Example5.2Solution:Solution:

Exercise5.1Considerasampleofmeasurement100withmean121cmandstandarddeviation7cmdrawnfromanormalpopulation.Trytocomputeitsstandarderror.Exercise5.1Solution:Solution:Section2

tdistributionSection2

tdistribution1.Definition

N(μ,

2)N(0,1)1.DefinitionN(μ,2)N(0,RandomsamplingRandomsamplingUsuallystandarddeviationσisunknown,sowecanonlygets,thenwecancalculateUsuallystandarddeviationσiThissamplingdistributionwasdevelopedbyW.SGossettandpublishedunderthepseudonym“student”in1908.itis,therefore,sometimescalledthe“student’stdistributionandisreallyafamilyofdistributionsdependentonthen-1.Thissamplingdistribution

＝n-1Zdistributiontdistribution＝n-1Zdistributiontdistribu2.thecharacteristicsoftdistributiongraphFIG4thegraphoftdistributionwithdifferentdegreesoffreedom2.thecharacteristicsoftdi1symmetricabout0;2theshapeoftcurveisdeterminedbydegreeoffreedom,df=n-1.3t-distributionisapproximatedtostandardnormaldistributionwhennisinfinite.

1symmetricabout0;總體特征抽樣調查的設計與分析課件tcriticalvaluewithone-sidedprobability→t(α,

)tcriticalvaluewithtwo-sidedprobability→t(α/2,

)tcriticalvaluewithone-sideExample5.2Withn=15,findt0suchthatP(-t0≤t≤

t0)=0.90Example5.2Withn=15,findsolutionFromtvaluetable,df=15-1=14,thetwo-tailedshadedareaequals0.10,so

-t0=-1.761and

t0=1.761solutionFromtvaluetablSection3confidenceintervalsforthepopulationmeanSection3StatisticalmethodsdescriptivestatisticsinferentialstatisticsparameterestimationhypothesistestIntervalsestimationPointestimationStatisticalmethodsdescriptive1.Basicconcepts

Parameterestimation:Deducethepopulationparameterbasingonthesamplestatistics1.BasicconceptsPointEstimateAsingle-valuedestimate.Asingleelementchosenfromasamplingdistribution.Conveyslittleinformationabouttheactualvalueofthepopulationparameterabouttheaccuracyoftheestimate.PointEstimateConfidenceIntervalorIntervalEstimationAnintervalorrangeofvaluesbelievedtoincludetheunknownpopulationparameter.ConfidenceIntervalorIntervaPointestimationLowerlimitUpperlimitIntervalsestimationPointestimationLowerlimitUpp1-aa/2a/21-aa/2a/2

2.MethodsZdistribution1.σ

isknown2.σ

isunknown，n＞50

tdistributionσ

isunknown，n≤50CICI2.MethodsZdistribution1.σExample5.3

Ahorticulturalscientistisdevelopinganewvarietyofapple.Oneoftheimportanttraits,inadditiontotaste,color,andstorability,istheuniformityofthefruitsize.Toestimatetheweightshesamples100maturefruitandcalculatesasamplemeanof220gandstandarddeviation5gDevelop95%confidenceintervalsforthepopulationmeanμfromhersampleExample5.3solution95%confidenceintervalsforthepopulationmeanisbetween219.02and220.98gsolution95%confidenceinteExerciseAforesterisinterestedinestimatingtheaveragenumberof‘counttrees’peracre.Arandomsampleofn=64oneacreisselectedandexamined.Theaverage(mean)numberofcounttreesperacreisfoundtobe27.3,withastandarddeviationof12.1.Usethisinformationtoconstruct95%confidenceintervalforμ.Exercisesolution95%confidenceintervalsforthepopulationmeanisbetween24.36and30.24solution95%confidenceinteTheforesteris95%confidentthatthepopulationmeanfor“counttrees”peracreisbetween24.36and30.24Theforesteris95%confidenExample5.4Theecologistsamples25plantsandmeasurestheirheights.Hefindsthatthesamplehasameanof15cmandasampledeviationof4cm.whatisthe95%confidenceintervalforthepopulationmeanμExample5.4solutiondf=25-1=24solutiondf=25-1=24Theplantecologistis95%confidentthatthepopulationmeanforheightsoftheseplantsisbetween13.349and16.65