CHAPTER-6

SamplingerrorandconfidenceintervalspopulationsamplestatisticParametererrorSection1samplingerrorofmeanSection2tdistributionSection3confidenceintervalsforthepopulationmeanSection1

samplingerrorofmean

AsimplerandomsampleisasampleofsizendrawnfromapopulationofsizeNinsuchawaythateverypossiblerandomsamplesnhasthesameprobabilityofbeingselected.Variabilityamongthesimplerandomsamplesdrawnfromthesamepopulationiscalledsamplingvariability,andtheprobabilitydistributionthatcharacterizessomeaspectofthesamplingvariability,usuallythemeanbutnotalways,iscalledasamplingdistribution.Thesesamplingdistributionsallowustomakeobjectivestatementsaboutpopulationparameterswithoutmeasuringeveryobjectinthepopulation.[Example1]ThepopulationmeanofDBPintheChineseadultmenis72mmHgwithstandarddeviation5mmHg.10adultparticipantswaschosenrandomlyfromtheChineseadultmen,herewecancalculatethesamplemeanandsamplestandarddeviation.Supposingsampling100times,what’stheresult?linkageNIfrandomsamplesarerepeatedlydrawnfromapopulationwithameanμandstandarddeviationσ,wecanfind:1thesamplemeansaredifferentfromtheothers2Thesamplemeanarenotnecessaryequaltopopulationmeanμ3ThedistributionofsamplemeanissymmetricaboutμHOWTOEXPLORETHESAMPLINGDISTRIBUTIONFORTHEMEAN?Thedifferencebetweensamplestatisticsandpopulationparameterorthedifferenceamongsamplestatisticsarecalledsamplingerror.Inreallifewesampleonlyonce,butwerealizethatoursamplecomesfromatheoreticalsamplingdistributionofallpossiblesamplesofaparticularsize.Thesamplingdistributionconceptprovidesalinkbetweensamplingvariabilityandprobability.Choosingarandomsampleisachanceoperationandgeneratingthesamplingdistributionconsistsofmanyrepetitionsofthischanceoperation.Whensamplingfromanormallydistributedpopulationwithmeanμ,thedistributionofthesamplemeanwillbenormalwithmeanμCentrallimitTheorem

=50

=10XPopulationdistributionn=4SamplingdistributionXn=16Whensamplingfromanonnormallydistributedpopulationwithmeanμ,thedistributionofthesamplemeanwillbeapproximatelynormalwithmeanμaslongasnislargerenough(n>50).CentrallimitTheoremXStandarderror(SE)canbeusedtoassesssamplingerrorofmean.Althoughsamplingerrorisinevitable,itcanbecalculatedaccurately.theoreticalvalueofSEestimationofSECalculationofstandarderror(SE)s↑→SE↑n↑→SE↓linkageExample5.2Oneanalystchoserandomlyasample(n=100)andmeasuredtheirweightswithameanof72kgandstandarddeviationof15kg.Question:whatisthestandarderror?Solution:

Exercise5.1Considerasampleofmeasurement100withmean121cmandstandarddeviation7cmdrawnfromanormalpopulation.Trytocomputeitsstandarderror.Solution:Section2

tdistribution1.Definition

N(μ,

2)N(0,1)RandomsamplingUsuallystandarddeviationσisunknown,sowecanonlygets,thenwecancalculateThissamplingdistributionwasdevelopedbyW.SGossettandpublishedunderthepseudonym“student”in1908.itis,therefore,sometimescalledthe“student’stdistributionandisreallyafamilyofdistributionsdependentonthen-1.

＝n-1Zdistributiontdistribution2.thecharacteristicsoftdistributiongraphFIG4thegraphoftdistributionwithdifferentdegreesoffreedom1symmetricabout0;2theshapeoftcurveisdeterminedbydegreeoffreedom,df=n-1.3t-distributionisapproximatedtostandardnormaldistributionwhennisinfinite.

tcriticalvaluewithone-sidedprobability→t(α,

)tcriticalvaluewithtwo-sidedprobability→t(α/2,

)Example5.2Withn=15,findt0suchthatP(-t0≤t≤

t0)=0.90solutionFromtvaluetable,df=15-1=14,thetwo-tailedshadedareaequals0.10,so

-t0=-1.761and

t0=1.761Section3confidenceintervalsforthepopulationmeanStatisticalmethodsdescriptivestatisticsinferentialstatisticsparameterestimationhypothesistestIntervalsestimationPointestimation1.Basicconcepts

Parameterestimation:DeducethepopulationparameterbasingonthesamplestatisticsPointEstimateAsingle-valuedestimate.Asingleelementchosenfromasamplingdistribution.Conveyslittleinformationabouttheactualvalueofthepopulationparameterabouttheaccuracyoftheestimate.ConfidenceIntervalorIntervalEstimationAnintervalorrangeofvaluesbelievedtoincludetheunknownpopulationparameter.PointestimationLowerlimitUpperlimitIntervalsestimation1-aa/2a/2

2.MethodsZdistribution1.σ

isknown2.σ

isunknown，n＞50

tdistributionσ

isunknown，n≤50CICIExample5.3

Ahorticulturalscientistisdevelopinganewvarietyofapple.Oneoftheimportanttraits,inadditiontotaste,color,andstorability,istheuniformityofthefruitsize.Toestimatetheweightshesamples100maturefruitandcalculatesasamplemeanof220gandstandarddeviation5gDevelop95%confidenceintervalsforthepopulationmeanμfromhersamplesolution95%confidenceintervalsforthepopulationmeanisbetween219.02and220.98gExerciseAforesterisinterestedinestimatingtheaveragenumberof‘counttrees’peracre.Arandomsampleofn=64oneacreisselectedandexamined.Theaverage(mean)numberofcounttreesperacreisfoundtobe27.3,withastandarddeviationof12.1.Usethisinformationtoconstruct95%confidenceintervalforμ.solution95%confidenceintervalsforthepopulationmeanisbetween24.36and30.24Theforesteris95%confidentthatthepopulationmeanfor“counttrees”peracreisbetween24.36and30.24Example5.4Theecologistsamples25plantsandmeasurestheirheights.Hefindsthatthesamplehasameanof15cmandasampledeviationof4cm.whatisthe95%confidenceintervalforthepopulationmeanμsolutiondf=25-1=24Theplantecologistis95%confidentthatthepopulationmeanforheig