Download Section 6-4 EXAMPLE - Gordon State College

2/3/2017
Section 6-4
Sampling Distributions
and Estimators
EXAMPLE
Becauseofrudesalespersonnel,apoor
businessplan,ineffectiveadvertising,andapoor
name,PollyEsther’sFashionswasinbusiness
onlythreedays.Onthefirstday1dresswas
sold,2weresoldonthesecondday,andonly5
weresoldonthethirdday.Because1,2,and5
aretheentirepopulation,themeanis
2.7
andthestandarddeviationis
1.7.
Let’sconsidersamplesofsize2.Thereare
only9differentpossiblesamplesofsize2,
assumingwesamplewithreplacement.
WHY SAMPLE WITH
REPLACEMENT?
1. Whenselectingarelativelysmallsample
fromalargepopulation,itmakesno
significantdifferencewhetherwesample
withreplacementorwithoutreplacement.
2. Samplingwithreplacementresultsin
independenteventsthatareunaffectedby
previousoutcomes,andindependent
eventsareeasiertoanalyzeandthey
resultinsimplerformulas.
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SAMPLING DISTRIBUTION OF A
STATISTIC
Thesamplingdistributionofastatistic
(suchasasamplemeanorsampleproportion)
isthedistributionofallvaluesofthestatistic
whenallpossiblesamplesofthesamesize
aretakenfromthesamepopulation.
Thesamplingdistributionofastatisticis
typicallyrepresentedasaprobability
distributionintheformofatable,probability
histogram,orformula.
SAMPLING DISTRIBUTION OF
THE SAMPLE MEAN
Thesamplingdistributionofthesample
mean isthedistributionofallpossiblesample
means(orthedistributionofthevariable ̅ ),
withallsampleshavingthesamesize taken
fromthesamepopulation.
Thesamplingdistributionofthesamplemean
istypicallyrepresentedasaprobability
distributionintheformofatable,probability
histogram,orformula.
SAMPLING VARIABILITY
Thevalueofastatistic,suchasthesample
mean ̅ ,dependsontheparticularvalues
includedinthesample,anditgenerallyvaries
fromsampletosample.Thisvariabilityofa
statisticiscalledsamplingvariability.
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SAMPLING DISTRIBUTION OF
THE SAMPLE VARIANCE
Thesamplingdistributionofthesample
variance isthedistributionofallpossible
samplevariances(orthedistributionofthe
variable ),withallsampleshavingthesame
size takenfromthesamepopulation.
Thesamplingdistributionofthesample
varianceistypicallyrepresentedasa
probabilitydistributionintheformofatable,
probabilityhistogram,orformula.
SAMPLING DISTRIBUTION OF
THE PROPORTION
Thesamplingdistributionofthesample
proportion isthedistributionofsample
proportions,withallsampleshavingthesame
samplesize .
NotationforProportions:
population proportion
sample proportion
Thenextslideshowsthesampling
distributionsofseveralstatisticsforour
exampleaswellasthesamplingdistribution
oftheproportionofoddnumbersforour
example.
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PROPERTIES OF THE DISTRIBUTION
OF SAMPLE MEANS
• Thesamplemeanstargetthevalueofthe
populationmean.(Thatis,themeanofthe
samplemeansisthepopulationmean.The
expectedvalueofthesamplemeansisequalto
thepopulationmean.)
• Thedistributionofsamplemeanstendstobea
normaldistribution.(Thiswillbediscussed
furtherinthenextsection,butthedistribution
tendstobecomeclosertoanormal
distributionasthesamplesizeincrease.)
PROPERTIES OF THE DISTRIBUTION
OF SAMPLE VARIANCES
• Thesamplevariancestendtotargetthe
valueofthepopulationvariance.(Thatis,
themeanofthesamplevariancesisthe
populationvariance.Theexpectedvalueof
thesamplevarianceisequaltothe
populationvariance.)
• Thedistributionofsamplevariancestends
tobeadistributionskewedtotheright.
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PROPERTIES OF THE DISTRIBUTION
OF SAMPLE PROPORTIONS
• Thesampleproportionstendtotargetthe
valueofthepopulationproportion.(That
is,themeanofthesampleproportionsis
thepopulationproportion.Theexpected
valueofthesampleproportionsisequalto
thepopulationproportion.)
• Thedistributionofsampleproportions
tendstobeanormaldistribution.
ESTIMATORS
Anestimator isastatisticthatisusedtoinfer
(estimate)thevalueofapopulationparameter.
Anunbiasedestimator isastatisticthat
targetsthevalueofthepopulationparameter
inthesensethatthesamplingdistributionof
thestatistichasameanthatisequaltothe
meanofthecorrespondingparameter.
BIASED AND UNBIASED
ESTIMATORS
• YouwillnoticefromTable6‐7thatsome
samplestatistics(themean,variance,and
proportion)“target”thepopulation
parameters.Thesesamplestatisticsare
calledunbiasedestimators.
• Othersamplestatistics(themedian,range,
andstandarddeviation)eitheroverestimate
orunderestimatethepopulationparameter.
Thesesamplestatisticsarecalledbiased
estimators.
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A COMMENT ON THE
STANDARD DEVIATION
Eventhoughthestandarddeviationisbiased,
thebiasisrelativelysmallinalargesample.
Asaresults isoftenusedtoestimateσ.
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