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SamplingMethodsandtheCentralLimitTheorem

Dr.RichardJerz

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GOALS

• Explainwhyasampleistheonlyfeasiblewaytolearnaboutapopulation.

• Describemethodstoselectasample.• Defineandconstructasamplingdistributionofthesamplemean.

• Explainthecentrallimittheorem.• UsetheCentralLimitTheoremtofindprobabilitiesofselectingpossiblesamplemeansfromaspecifiedpopulation.

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WhySamplethePopulation?

• Thephysicalimpossibilityofcheckingallitemsinthepopulation.

• Thecost ofstudyingalltheitemsinapopulation.

• Contactingthewholepopulationwouldoftenbetime-consuming.

• Thedestructive natureofcertaintests.• Thesampleresultsareusuallyadequate.

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ProbabilitySampling

• Aprobabilitysampleisasampleselectedsuchthateachitemorpersoninthepopulationbeingstudiedhasaknownlikelihoodofbeingincludedinthesample.

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4MethodsofProbabilitySampling

1. SimpleRandomSample:Asampleformulatedsothateachitemorpersoninthepopulationhasthesamechanceofbeingincluded.(userandomnumbers)

2. SystematicRandomSampling:Theitemsorindividualsofthepopulationarearrangedinsomeorder.Arandomstartingpoint(randomnumber)isselectedandtheneverykth memberofthepopulationisselectedforthesample.

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MethodsofProbabilitySampling

3. StratifiedRandomSampling:Apopulationisfirstdividedintosubgroups,calledstrata,andasampleisrandomlyselectedfromeachstratum.Example:men&women

4. ClusterSampling:Apopulationisfirstdividedintoprimary(geographic)unitsthensamplesareselectedfromtheprimaryunits.

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ProducingRandomNumber

• MSExcel

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SamplingError

• Thesamplingerroristhedifferencebetweenasamplestatisticanditscorrespondingpopulationparameter.

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SampleMeans

• Thesamplingdistributionofthesamplemeanisaprobabilitydistributionconsistingofallpossiblesamplemeansofagivensamplesizeselectedfromapopulation.

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Example:SamplingDistributionofSampleMeans

• Tartus Industrieshassevenproductionemployees(consideredthepopulation).Thehourlyearningsofeachemployeearegiveninthetablebelow.

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1.Whatisthepopulationmean?2.Whatisthesamplingdistributionofthesamplemeanforsamplesofsize2?3.Whatisthemeanofthesamplingdistribution?4.Whatobservationscanbemadeaboutthepopulationandthesamplingdistribution?

SamplingDistributionoftheSampleMeans– Example

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Example:SamplingDistributionofSampleMeans

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ExampleSamplingDistributionofSampleMeans

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Sampling

• Largersamplesarebetterthansmaller(Excel)

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CentralLimitTheorem

• Forapopulationwithameanμ andavarianceσ2 thesamplingdistributionofthemeansofallpossiblesamplesofsizengeneratedfromthepopulationwillbeapproximatelynormallydistributed.(ExcelModel)

• Themeanofthesamplingdistributionequaltoμ andthevarianceequaltoσ2/n.

• The“standarderrorofthemean”isσ/√n

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UsingtheSamplingDistributionoftheSample

Mean(SigmaKnown)• Ifapopulationfollowsthenormaldistribution,thesamplingdistributionofthesamplemeanwillalsofollowthenormaldistribution.

• Todeterminetheprobabilityasamplemeanfallswithinaparticularregion,use:

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nXzs

µ-=

Example(SigmaKnown)TheQualityAssuranceDepartmentforCola,Inc.,maintains

recordsregardingtheamountofcolainitsJumbobottle.Theactualamountofcolaineachbottleiscritical,butvariesasmallamountfromonebottletothenext.Cola,Inc.,doesnotwishtounderfillthebottles.Ontheotherhand,itcannotoverfilleachbottle.Itsrecordsindicatethattheamountofcolafollowsthenormalprobabilitydistribution.Themeanamountperbottleis31.2ouncesandthepopulationstandarddeviationis0.4ounces.At8A.M.todaythequalitytechnicianrandomlyselected16bottlesfromthefillingline.Themeanamountofcolacontainedinthebottlesis31.38ounces.

Isthisanunlikelyresult?Isitlikelytheprocessisputtingtoomuchsodainthebottles?Toputitanotherway,isthesamplingerrorof0.18ouncesunusual?

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TheSamplingDistributionoftheSampleMean

Step1:Findthez-valuescorrespondingtothesamplemeanof31.38

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80.1164.0$20.3138.31

=-

=-

=n

Xzs

µ

TheSamplingDistributionoftheSampleMean

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Step2:FindtheprobabilityofobservingaZequaltoorgreaterthan1.80

TheSamplingDistributionoftheSampleMean

Whatdoweconclude?

Itisunlikely,lessthana4percentchance,wecouldselectasampleof16observationsfromanormalpopulationwithameanof31.2ouncesandapopulationstandarddeviationof0.4ouncesandfindthesamplemeanequaltoorgreaterthan31.38ounces.

Weconcludetheprocessisputtingtoomuchcolainthebottles.

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