BasicBusinessStatistics,11e?2009Prentice-Hall,Inc.Chap7-1Chapter7SamplingandSamplingDistributionsBusinessStatistics:AFirstCourse

5thEditionBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-2LearningObjectivesInthischapter,youlearn:

TodistinguishbetweendifferentsamplingmethodsTheconceptofthesamplingdistributionTocomputeprobabilitiesrelatedtothesamplemeanandthesampleproportionTheimportanceoftheCentralLimitTheoremBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-3WhySample?Selectingasampleislesstime-consumingthanselectingeveryiteminthepopulation(census).Selectingasampleislesscostlythanselectingeveryiteminthepopulation.Ananalysisofasampleislesscumbersomeandmorepracticalthanananalysisoftheentirepopulation.BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-4ASamplingProcessBeginsWithASamplingFrameThesamplingframeisalistingofitemsthatmakeupthepopulationFramesaredatasourcessuchaspopulationlists,directories,ormapsInaccurateorbiasedresultscanresultifaframeexcludescertainportionsofthepopulationUsingdifferentframestogeneratedatacanleadtodissimilarconclusionsBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-5TypesofSamplesSamplesNon-ProbabilitySamplesJudgmentProbabilitySamplesSimpleRandomSystematicStratifiedClusterConvenienceBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-6TypesofSamples:

NonprobabilitySampleInanonprobabilitysample,itemsincludedarechosenwithoutregardtotheirprobabilityofoccurrence.Inconveniencesampling,itemsareselectedbasedonlyonthefactthattheyareeasy,inexpensive,orconvenienttosample.Inajudgmentsample,yougettheopinionsofpre-selectedexpertsinthesubjectmatter.

BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-7TypesofSamples:

ProbabilitySampleInaprobabilitysample,itemsinthesamplearechosenonthebasisofknownprobabilities.ProbabilitySamplesSimpleRandomSystematicStratifiedClusterBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-8ProbabilitySample:

SimpleRandomSampleEveryindividualoritemfromtheframehasanequalchanceofbeingselectedSelectionmaybewithreplacement(selectedindividualisreturnedtoframeforpossiblereselection)orwithoutreplacement(selectedindividualisn’treturnedtotheframe).Samplesobtainedfromtableofrandomnumbersorcomputerrandomnumbergenerators.BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-9SelectingaSimpleRandomSampleUsingARandomNumberTableSamplingFrameForPopulationWith850ItemsItemNameItem#BevR. 001UlanX. 002. .. .. .. .JoannP. 849PaulF. 850PortionOfARandomNumberTable492808892435779002838116307275111000234012860746979664489439098932399720048494208887208401TheFirst5ItemsinasimplerandomsampleItem#492Item#808Item#892--doesnotexistsoignoreItem#435Item#779Item#002BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-10Decideonsamplesize:nDivideframeofNindividualsintogroupsofkindividuals:k=N/nRandomlyselectoneindividualfromthe1stgroupSelecteverykthindividualthereafterProbabilitySample:

SystematicSampleN=40n=4k=10FirstGroupBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-11BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-12ProbabilitySample:

StratifiedSampleDividepopulationintotwoormoresubgroups(calledstrata)accordingtosomecommoncharacteristicAsimplerandomsampleisselectedfromeachsubgroup,withsamplesizesproportionaltostratasizesSamplesfromsubgroupsarecombinedintooneThisisacommontechniquewhensamplingpopulationofvoters,stratifyingacrossracialorsocio-economiclines.PopulationDividedinto4strataBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-13ProbabilitySample

ClusterSamplePopulationisdividedintoseveral“clusters,”eachrepresentativeofthepopulationAsimplerandomsampleofclustersisselectedAllitemsintheselectedclusterscanbeused,oritemscanbechosenfromaclusterusinganotherprobabilitysamplingtechniqueAcommonapplicationofclustersamplinginvolveselectionexitpolls,wherecertainelectiondistrictsareselectedandsampled.Populationdividedinto16clusters.RandomlyselectedclustersforsampleBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-14ProbabilitySample:

ComparingSamplingMethodsSimplerandomsampleandSystematicsampleSimpletouseMaynotbeagoodrepresentationofthepopulation’sunderlyingcharacteristicsStratifiedsampleEnsuresrepresentationofindividualsacrosstheentirepopulationClustersampleMorecosteffectiveLessefficient(needlargersampletoacquirethesamelevelofprecision)BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-15EvaluatingSurveyWorthinessWhatisthepurposeofthesurvey?Isthesurveybasedonaprobabilitysample?Coverageerror–appropriateframe?Nonresponseerror–followupMeasurementerror–goodquestionselicitgoodresponsesSamplingerror–alwaysexistsBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-16TypesofSurveyErrorsCoverageerrororselectionbiasExistsifsomegroupsareexcludedfromtheframeandhavenochanceofbeingselectedNonresponseerrororbiasPeoplewhodonotrespondmaybedifferentfromthosewhodorespondSamplingerrorVariationfromsampletosamplewillalwaysexistMeasurementerrorDuetoweaknessesinquestiondesign,respondenterror,andinterviewer’seffectsontherespondent(“Hawthorneeffect”)BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-17TypesofSurveyErrorsCoverageerrorNonresponseerrorSamplingerrorMeasurementerrorExcludedfromframeFollowuponnonresponsesRandomdifferencesfromsampletosampleBadorleadingquestion(continued)BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-18SamplingDistributionsAsamplingdistributionisadistributionofallofthepossiblevaluesofasamplestatisticforagivensizesampleselectedfromapopulation.Forexample,supposeyousample50studentsfromyourcollegeregardingtheirmeanGPA.Ifyouobtainedmanydifferentsamplesof50,youwillcomputeadifferentmeanforeachsample.WeareinterestedinthedistributionofallpotentialmeanGPAwemightcalculateforanygivensampleof50students.BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-19Developinga

SamplingDistributionAssumethereisapopulation…PopulationsizeN=4Randomvariable,X,

isageofindividualsValuesofX:18,20,

22,24(years)ABCDBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-0

18202224

ABCDUniformDistributionP(x)x(continued)SummaryMeasuresforthePopulationDistribution:Developinga

SamplingDistributionBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-2116possiblesamples(samplingwithreplacement)Nowconsiderallpossiblesamplesofsizen=2(continued)Developinga

SamplingDistribution16SampleMeans1stObs2ndObservation182022241818,1818,2018,2218,242020,1820,2020,2220,242222,1822,2022,2222,242424,1824,2024,2224,24BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-22SamplingDistributionofAllSampleMeans181920212223P(X)

XSampleMeansDistribution16SampleMeans_DevelopingaSamplingDistribution(continued)(nolongeruniform)_BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-23SummaryMeasuresofthisSamplingDistribution:DevelopingaSamplingDistribution(continued)BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-24ComparingthePopulationDistribution

totheSampleMeansDistribution181920212223P(X)X

18

20

22

24AB

C

DPopulationN=4P(X)X_SampleMeansDistributionn=2_BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-25SampleMeanSamplingDistribution:

StandardErroroftheMeanDifferentsamplesofthesamesizefromthesamepopulationwillyielddifferentsamplemeansAmeasureofthevariabilityinthemeanfromsampletosampleisgivenbytheStandardErroroftheMean:

(Thisassumesthatsamplingiswithreplacementor samplingiswithoutreplacementfromaninfinitepopulation)NotethatthestandarderrorofthemeandecreasesasthesamplesizeincreasesBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-26SampleMeanSamplingDistribution:

IfthePopulationisNormalIfapopulationisnormallydistributedwithmeanμandstandarddeviationσ,thesamplingdistributionofisalsonormallydistributedwith and

BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-27Z-valueforSamplingDistribution

oftheMeanZ-valueforthesamplingdistributionof:where: =samplemean =populationmean =populationstandarddeviation

n=samplesizeBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-28NormalPopulationDistributionNormalSamplingDistribution(hasthesamemean)SamplingDistributionProperties

(i.e.isunbiased

)BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-29SamplingDistributionPropertiesAsnincreases,decreasesLargersamplesizeSmallersamplesize(continued)BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-30DeterminingAnIntervalIncludingAFixedProportionoftheSampleMeansFindasymmetricallydistributedintervalaroundμthatwillinclude95%ofthesamplemeanswhenμ=368,σ=15,andn=25.Sincetheintervalcontains95%ofthesamplemeans5%ofthesamplemeanswillbeoutsidetheintervalSincetheintervalissymmetric2.5%willbeabovetheupperlimitand2.5%willbebelowthelowerlimit.Fromthestandardizednormaltable,theZscorewith2.5%(0.0250)belowitis-1.96andtheZscorewith2.5%(0.0250)aboveitis1.96.BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-31DeterminingAnIntervalIncludingAFixedProportionoftheSampleMeansCalculatingthelowerlimitoftheintervalCalculatingtheupperlimitoftheinterval95%ofallsamplemeansofsamplesize25arebetween362.12and373.88(continued)BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-32SampleMeanSamplingDistribution:

IfthePopulationisnotNormalWecanapplytheCentralLimitTheorem:Evenifthepopulationisnotnormal,…samplemeansfromthepopulationwillbe

approximatelynormalaslongasthesamplesizeislargeenough.Propertiesofthesamplingdistribution:andBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-33n↑CentralLimitTheoremAsthesamplesizegetslargeenough…thesamplingdistributionbecomesalmostnormalregardlessofshapeofpopulationBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-34PopulationDistributionSamplingDistribution(becomesnormalasnincreases)CentralTendencyVariationLargersamplesizeSmallersamplesizeSampleMeanSamplingDistribution:

IfthePopulationisnotNormal(continued)Samplingdistributionproperties:BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-35HowLargeisLargeEnough?Formostdistributions,n>30willgiveasamplingdistributionthatisnearlynormalForfairlysymmetricdistributions,n>15willusuallygiveasamplingdistributionisalmostnormalFornormalpopulationdistributions,thesamplingdistributionofthemeanisalwaysnormallydistributedBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-36ExampleSupposeapopulationhasmeanμ=8andstandarddeviationσ=3.Supposearandomsampleofsizen=36isselected.Whatistheprobabilitythatthesamplemeanisbetween7.8and8.2?BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-37ExampleSolution:Evenifthepopulationisnotnormallydistributed,thecentrallimittheoremcanbeused(n>30)…sothesamplingdistributionofisapproximatelynormal…withmean=8…andstandarddeviation

(continued)BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-38ExampleSolution(continued):(continued)Z7.88.2-0.40.4SamplingDistributionStandardNormalDistribution.1554+.1554PopulationDistribution????????????SampleStandardizeXBasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-39PopulationProportions

π=theproportionofthepopulationhavingsomecharacteristicSampleproportion

(p)providesanestimateofπ:0≤p≤1pisapproximatelydistributedasanormaldistributionwhennislarge (assumingsamplingwithreplacementfromafinitepopulationorwithoutreplacementfromaninfinitepopulation)BasicBusinessStatistics,11e?2009Prentice-Hall,Inc..Chap7-40SamplingDistributionofpApproximatedbya

normaldistributionif:

where

and(whereπ=populationproportion)SamplingDistributi