Notes 07Sampling MethodsCLT.docx
- 文档编号:15802991
- 上传时间:2023-07-08
- 格式:DOCX
- 页数:15
- 大小:96.57KB
Notes 07Sampling MethodsCLT.docx
《Notes 07Sampling MethodsCLT.docx》由会员分享,可在线阅读,更多相关《Notes 07Sampling MethodsCLT.docx(15页珍藏版)》请在冰点文库上搜索。
Notes07SamplingMethodsCLT
Chapter8
SAMPLINGMETHODS&CENTRALLIMITTHEOREM
Outline
1ReviewonProbabilityDistributions
2SamplingDistribution
3SamplingDistributionforSampleMeans
4SamplingDistributionforSampleProportions
OBJECTIVES
Bytheendofthischapter,youwillbeableto:
1.Definethesamplingprocess.
2.Distinguishbetweenpopulation,sampleandsamplingdistributions.
3.Describethecharacteristicsofsamplingdistributionofthemean.
4.StatetheCentralLimitTheorem.
5.ApplytheCentralLimitTheorem.
6.Convertsamplingvariabletostandardizedvariableandviceversa.
7.Describethecharacteristicsofthesamplingdistributionofproportion.
1REVIEW
ProbabilityDistribution
AprobabilitydistributionisalistoftheprobabilitiesofALLthepossibleoutcomeofavariableinasurvey/experiment.
TypesofProbabilityDistributions
(a)PopulationDistribution
Itistheprobabilitydistributionofallvaluesofavariableinapopulation.
Populationdistributionsaredescribedbyparameters.
e.g.populationmean,populationstandarddeviation,andpopulationproportion.
(b)SampleDistribution
Itistheprobabilitydistributionofallvaluesofavariableinasample.
Sampledistributionsaredescribedbysamplestatistics.
e.g.samplemean,samplestandarddeviation,sampleproportion.
Samplestatisticsenableustomakeinferences,predictionsanddecisionsabouttheparameters.
2SamplingDistribution
AsamplingdistributionisaprobabilitydistributionofALLpossiblesamplestatisticvaluesofagivensamplesize.
TypesofSamplingDistributions
∙Samplingdistributionforthesamplemeans
∙Samplingdistributionforthesampleproportions
P
Samplingdistributionforsampleproportions
Aprobabilitydistributionofallpossiblesampleproportionvaluesofagivensamplesize
Samplingdistributionforsamplemeans
Aprobabilitydistributionofallpossiblesamplemeanvaluesofagivensamplesize
3Samplingdistributionforsamplemeans
Itisaprobabilitydistributionofallpossiblesamplemeanvaluesofagivensamplesize.
3.1DescribingtheSamplingDistributionoftheSampleMean
Thetwomajorcharacteristicsdescribingthesamplingdistributionare:
Meanofthesamplingdistribution
Thisistheaveragevalueofallthestatisticvalues.(e.g.allsamplemeanvalues)
themeanofthesamplingdistributionforthesamplemeansisdenotedby
Ingeneral,themeanofthesamplingdistributionequalsthepopulationmean,i.e.
=
(pleaserefertotheexampleinpg259–261inyourtextbook)
StandardErrorofthesamplingdistribution.
Thisisthestandarddeviationofallthestatisticvalues.
Thestandarderrorforthesamplemeanisdenotedas
:
Formula
ifNisinfinite;orifNisfiniteandn/N0.05,then
ConceptsofStandardError
SamplingError
Itisunlikelythatthesamplemeanwouldbeidenticaltothepopulationmean.Weshouldexpectsomedifferencebetweenthesamplemeanandthepopulation.Thisdifferenceiscalledsamplingerror.So,
Samplingerror=Thedifferencebetweeneachsamplestatisticvalueandthepopulationparametervalue.i.e.|
|
StandardError
-Theaveragesizeofthesamplingerrorinasamplingdistribution.
-Thestandarddeviationofallthesamplemeans.
-Itmeasuresthespreadofthesamplingdistributionforthesamplemeans.
-Itissmallerthanthepopulationstandarddeviation.Thelargerthesamplesize,thesmallerthestandarderror.
3.2DecidingtheShapeoftheSamplingDistributionoftheSampleMean
Therearemanytypesofdistributionasamplingdistributioncanfollow:
e.g.anormaldistribution,at-distribution,fdistributionor2distribution.Forthislecture,wewilllearntofindprobabilitiesusingasamplingdistributionthatfollowsanormaldistribution.
(a)SamplingfromaNormallyDistributedPopulationwithKnownstandarddeviation
Ifwesamplefromapopulationthatisnormallydistributedwithmeanandstandarddeviation,thesamplingdistributionofthemeanwillalsobenormallydistributedforanysamplesizen.
(b)SamplingfromNon-NormallyDistributedPopulation
LargeSampleSize
Ifthesampleisdrawnfromanon-normalpopulation,thenthesamplingdistributionisApproximatelynormallydistributed,isn30.(CentralLimitTheoremapplies).
CentralLimitTheorem
Ifallsamplesofaparticularsizeareselectedfromanypopulation,thesamplingdistributionofthesamplemeanisapproximatelyanormaldistribution.Thisapproximationimproveswithlargersamples.
Thisistrue,regardlessoftheshapeofthedistributionofthepopulation,butifapopulationdistributionisskewed,thenitmayrequiren>=30forthesamplingdistributiontoapproachnormaldistribution.
SignificanceoftheCentralLimitTheorem
ItpermitsustousesamplestatisticstomakeinferenceaboutthepopulationparametersWITHOUTknowingabouttheshapeofthepopulationdistribution.
3.3FindingprobabilitieswithaSamplingDistribution
Inordertofindprobabilitiesfromasamplingdistributionofthemeanthatisnormallydistribution,wewillhavetoconvertthesamplemeanvaluestotheStandardNormalvalue,Z.
StandardNormalValuesforsamplingdistribution
Z=
Example1
Thedistributionofannualearningsofallbanktellerswithfiveyears'experienceisskewednegativelywithmean$15,000andastandarddeviationof$2,000.Ifwedrawarandomsampleof30tellers,whatistheprobabilitythattheirearningswillaveragemorethan$15,750annually?
findP(
>15750)
Step1:
Determinethestandarderror:
Step2:
Determinethesamplingdistributionandthestandardizedvalue
Sincen=30,byCentralLimitTheorem,thesamplingdistributionisanormaldistribution.
Z==2.05
Step3:
Sketchadiagram
Step4:
Findtheprobability
P(
>15750)=P(Z>2.05)=
Step5:
Conclusion
Theprobabilitythatthemeanearningsofthe30tellersexceeding$15750is0.0202
Example2
SarjitSingh,theownerofachainof25clothingstores,hasbeenthinkingaboutwindinguphisbusinessbecauseitisnotasprofitableasbefore.Overthepastfewyears,themeannetincomeforthe25storeshasbeen$21,000withastandarddeviationof$3,400.Hewouldsellthebusinessiftheaverageofthefirst5storesauditedatyearendislessthan$20,000.WhatistheprobabilitythatSarjitwillsellout?
(Assumethattheprofitsfollowanormaldistribution).
X:
=annualprofitsoftheclothingstores
findP(
<20,000)
Step1:
Determinethestandarderror:
_____________finitemultiplier
Step2:
DeterminetheSamplingdistributionandthestandardizedvalue.
Thedistributionoftheprofitsoftheclothingstoresisanormaldistribution(populationdistributionisnormal)andthepopulationstandarddeviationisknown.Thus,thesamplingdistributionfollowsanormaldistribution.
Step3:
Sketchadiagram
Step4:
Findtheprobability
P(Z<-0.72)=
Step5:
Conclusion
TheprobabilitythatSarjitwillselloutis0.2358
4SamplingDistributionforSampleProportions
UseofSampleProportion
Whendealingwithacategorical(qualitative)variablewhereeachindividualitemisansweredaseither“YES”or“NO”,intermsofwhethertheypossessaparticularcharacteristic,andweareinterestedintheproportionoftheobservationshavinga“YES”answer.
Notation
p:
Proportionofapopulationwithacertaincharacteristic;
:
SampleProportionwithacertaincharacteristic
SamplingDistributionforSampleProportion
Itisasamplingdistributionforallsampleproportionvalues.
4.1DescribingtheSamplingDistributionoftheSampleProportion
MeanoftheSamplingDistributionforSampleProportion:
Itistheaverageofallsampleproportionvalues.Ingeneral,themeanofallsampleproportionvaluesisthesameasthepopulationproportion.i.e.
StandardError
or
if
4.2TheShapeoftheSamplingdistributionforsampleproportions
NormalDistribution,whennp5ANDnq5(whereq=1–p)
4.3FindingProbabilitieswithSamplingDistributionwithSampleProportion
Example3
Assume40%ofthe5,000studentsintheNationalStadiumaregirls.Whatistheprobabilitythatinasimplerandomsampleof150students,35%orlesswillbegirls?
P:
ProportionofgirlsintheStadium.FindP(
<0.35)
Step1:
Determinethestandarderror:
finitemultipliesnotappliedhere
Step2:
DeterminetheSamplingdistributionandthestandardizedvalue.
Asnp=>5nq=>5
Thesamplingdistributionforthesampleproportionsisanormaldistribution.
Step3:
Sketchadiagram
Step4:
Findtheprobability
P(
<0.35)=P(Z<)
Step5:
Conclusion
Thechanceforasamplecontaining35%girlsis0.1056
Summary
SampleStatistics
Meanofsamplingdistribution
wherex=valueofthevariable
wherex=numberofsuccesscases
StandardError
IfpopulationNisinfiniteor
use
IfpopulationNisfiniteand
thenuse
IfpopulationNisinfinite
or
IfpopulationNisfiniteand
thenuse
Samplingdistribution
Samplingdistributionisanormaldistributionif:
∙samplesizen30byCentralLimittheoremOR
∙populationdistributionisanormaldistributionandisknown
Samplingdistributionisanormaldistribution
when:
np5andn(1–p)5
StandardizedFormula
StandardNormalDistribution
0Z
Z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
.0000
.0040
.0080
.0120
.0160
.0199
.0239
.0279
.0319
.0359
0.1
.0398
.0438
.0478
.0517
.0557
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- Notes 07Sampling Methods CLT 07 Sampling
![提示](https://static.bingdoc.com/images/bang_tan.gif)