ch08 - sampling methods and central limit - sampling methods and central limit.pptx author rick jerz...
TRANSCRIPT
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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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