(1.26)-PowerPoint Ch 08商务统计课件

Chapter8

IntervalEstimationPopulationMean:sKnownPopulationMean:sUnknownDeterminingtheSampleSizePopulationProportionApointestimatorcannotbeexpectedtoprovidetheexactvalueofthepopulationparameter.Anintervalestimatecanbecomputedbyaddingandsubtractingamarginoferrortothepointestimate.

PointEstimate+/-MarginofErrorThepurposeofanintervalestimateistoprovideinformationabouthowclosethepointestimateistothevalueoftheparameter.MarginofErrorandtheIntervalEstimateThegeneralformofanintervalestimateofapopulationmeanis

MarginofErrorandtheIntervalEstimateIntervalEstimateofaPopulationMean:

sKnownInordertodevelopanintervalestimateofapopulationmean,themarginoferrormustbecomputedusingeither:thepopulationstandarddeviations,orthesamplestandarddeviationssisrarelyknownexactly,butoftenagoodestimatecanbeobtainedbasedonhistoricaldataorotherinformation.Werefertosuchcasesasthesknowncase.Thereisa1-

probabilitythatthevalueofasamplemeanwillprovideamarginoferroroforless.

/2

/21-

ofallvaluesSamplingdistributionofIntervalEstimateofaPopulationMean:

sKnown

/2

/21-

ofallvaluesSamplingdistributionof[--------------------------------------------------][--------------------------------------------------][--------------------------------------------------]intervaldoesnotincludemintervalincludesmintervalincludesmIntervalEstimateofaPopulationMean:

sKnownIntervalEstimateofmIntervalEstimateofaPopulationMean:

sKnownwhere:isthesamplemean 1-

istheconfidencecoefficient

z

/2isthezvalueprovidinganareaof

/2intheuppertailofthestandard normalprobabilitydistribution

sisthepopulationstandarddeviation

nisthesamplesizeIntervalEstimateofaPopulationMean:

sKnownValuesofza/2fortheMostCommonlyUsedConfidenceLevels90%.10.05.95001.645ConfidenceTableLevela

a/2Look-upAreaza/2

95%.05.025.97501.96099%.01.005.99502.576MeaningofConfidence

Because90%ofalltheintervalsconstructedusingwillcontainthepopulationmean,wesayweare90%confidentthattheintervalincludesthepopulationmeanm.

Wesaythatthisintervalhasbeenestablishedatthe90%confidencelevel.

Thevalue.90isreferredtoastheconfidence

coefficient.IntervalEstimateofaPopulationMean:

KnownExample:DiscountSounds DiscountSoundshas260retailoutletsthroughouttheUnitedStates.Thefirmisevaluatingapotentiallocationforanewoutlet,basedinpart,onthemeanannualincomeoftheindividualsinthemarketingareaofthenewlocation.Asampleofsizen=36wastaken;thesamplemeanincomeis$41,100.Thepopulationisnotbelievedtobehighlyskewed.Thepopulationstandarddeviationisestimatedtobe$4,500,andtheconfidencecoefficienttobeusedintheintervalestimateis.95.95%ofthesamplemeansthatcanbeobservedarewithin+1.96ofthepopulationmean

.Themarginoferroris:Thus,at95%confidence,themarginoferroris$1,470.IntervalEstimateofaPopulationMean:

KnownExample:DiscountSoundsIntervalestimateof

is:IntervalEstimateofaPopulationMean:

KnownWeare95%confidentthattheintervalcontainsthepopulationmean.$41,100+$1,470or$39,630to$42,570Example:DiscountSoundsIntervalEstimateofaPopulationMean:

Known90%3.2978.71to85.29ConfidenceMarginLevelofErrorIntervalEstimate95%3.9278.08to85.9299%5.1576.85to87.15Example:DiscountSoundsInordertohaveahigherdegreeofconfidence,themarginoferrorandthusthewidthoftheconfidenceintervalmustbelarger.IntervalEstimateofaPopulationMean:

sKnownAdequateSampleSizeInmostapplications,asamplesizeofn=30isadequate.Ifthepopulationdistributionishighlyskewedorcontainsoutliers,asamplesizeof50ormoreisrecommended.IntervalEstimateofaPopulationMean:

sKnownAdequateSampleSize(continued)Ifthepopulationisbelievedtobeatleastapproximatelynormal,asamplesizeoflessthan15canbeused.Ifthepopulationisnotnormallydistributedbutisroughlysymmetric,asamplesizeassmallas15willsuffice.IntervalEstimateofaPopulationMean:

sUnknownIfanestimateofthepopulationstandarddeviationscannotbedevelopedpriortosampling,weusethesamplestandarddeviationstoestimates.Thisisthesunknowncase.Inthiscase,theintervalestimateformisbasedonthetdistribution.(We’llassumefornowthatthepopulationisnormallydistributed.)WilliamGosset,writingunderthename“Student”,isthefounderofthetdistribution.tDistributionGossetwasanOxfordgraduateinmathematicsandworkedfortheGuinnessBreweryinDublin.Hedevelopedthetdistributionwhileworkingonsmall-scalematerialsandtemperatureexperiments.Thetdistributionisafamilyofsimilarprobabilitydistributions.tDistributionAspecifictdistributiondependsonaparameterknownasthedegreesoffreedom.Degreesoffreedomrefertothenumberofindependentpiecesofinformationthatgointothecomputationofs.tDistributionAtdistributionwithmoredegreesoffreedomhaslessdispersion.Asthedegreesoffreedomincreases,thedifferencebetweenthetdistributionandthestandardnormalprobabilitydistributionbecomessmallerandsmaller.tDistributionStandardnormaldistributiontdistribution(20degreesoffreedom)tdistribution(10degreesoffreedom)0z,tFormorethan100degreesoffreedom,thestandardnormalzvalueprovidesagoodapproximationtothetvalue.tDistributionThestandardnormalzvaluescanbefoundintheinfinitedegrees(

)rowofthetdistributiontable.tDistributionStandardnormalzvaluesIntervalEstimatewhere:1-

=theconfidencecoefficient

t

/2=thetvalueprovidinganareaof

/2 intheuppertailofatdistribution withn-1degreesoffreedom

s=thesamplestandarddeviationIntervalEstimateofaPopulationMean:

sUnknownAreporterforastudentnewspaperiswritinganarticleonthecostofoff-campushousing.Asampleof16efficiencyapartmentswithinahalf-mileofcampusresultedinasamplemeanof$750permonthandasamplestandarddeviationof$55.IntervalEstimateofaPopulationMean:

sUnknownExample:ApartmentRentsLetusprovidea95%confidenceintervalestimateofthemeanrentpermonthforthepopulationofefficiencyapartmentswithinahalf-mileofcampus.Wewillassumethispopulationtobenormallydistributed. At95%confidence,

=.05,and

/2=.025.Inthetdistributiontableweseethatt.025=2.131.t.025isbasedonn

-1=16-1=15degreesoffreedom.IntervalEstimateofaPopulationMean:

sUnknownWeare95%confidentthatthemeanrentpermonthforthepopulationofefficiencyapartmentswithinahalf-mileofcampusisbetween$720.70and$779.30.IntervalEstimateIntervalEstimateofaPopulationMean:

sUnknownMarginofErrorIntervalEstimateofaPopulationMean:

s

UnknownAdequateSampleSizeIfthepopulationdistributionishighlyskewedorcontainsoutliers,asamplesizeof50ormoreisrecommended.Inmostapplications,asamplesizeofn=30is

adequatewhenusingtheexpressiontodevelopanintervalestimateofapopulationmean.IntervalEstimateofaPopulationMean:

s

UnknownAdequateSampleSize(continued)Ifthepopulationisbelievedtobeatleastapproximatelynormal,asamplesizeoflessthan15canbeused.Ifthepopulationisnotnormallydistributedbutisroughlysymmetric,asamplesizeassmallas15willsuffice.SummaryofIntervalEstimationProceduresforaPopulationMeanCanthepopulationstandarddeviationsbeassumedknown?UseYesNoUsesKnownCasesUnknownCaseUsethesamplestandarddeviationstoestimatesLetE=thedesiredmarginoferror.

Eistheamountaddedtoandsubtractedfromthepointestimatetoobtainanintervalestimate.SampleSizeforanIntervalEstimate

ofaPopulationMeanIfadesiredmarginoferrorisselectedpriortosampling,thesamplesizenecessarytosatisfythemarginoferrorcanbedetermined.SampleSizeforanIntervalEstimate

ofaPopulationMeanMarginofError

NecessarySampleSize

SampleSizeforanIntervalEstimate

ofaPopulationMeanTheNecessarySampleSizeequationrequiresavalueforthepopulationstandarddeviations.Ifsisunknown,apreliminaryorplanningvalueforscanbeusedintheequation.1.Usetheestimateofthepopulationstandarddeviationcomputedinapreviousstudy.2.Useapilotstudytoselectapreliminarystudyandusethesamplestandarddeviationfromthestudy.3.Usejudgmentora“bestguess”forthevalueofs.

RecallthatDiscountSoundsisevaluatingapotentiallocationforanewretailoutlet,basedinpart,onthemeanannualincomeoftheindividualsinthemarketingareaofthenewlocation.SampleSizeforanIntervalEstimate

ofaPopulationMeanExample:DiscountSoundsSupposethatDiscountSounds’managementteamwantsanestimateofthepopulationmeansuchthatthereisa.95probabilitythatthesamplingerroris$500orless.Howlargeasamplesizeisneededtomeettherequiredprecision?

At95%confidence,z.025=1.96.Recallthat

=4,500.SampleSizeforanIntervalEstimate

ofaPopulationMeanAsampleofsize312isneededtoreachadesiredprecisionof+$500at95%confidence.Thegeneralformofanintervalestimateofapopulationproportionis

IntervalEstimate

ofaPopulationProportionIntervalEstimate

ofaPopulationProportionThesamplingdistributionofplaysakeyroleincomputingthemarginoferrorforthisintervalestimate.Thesamplingdistributionof

canbeapproximatedbyanormaldistributionwhenevernp

>5and

n(1–p)>5.

/2

/2IntervalEstimate

ofaPopulationProportionNormalApproximationofSamplingDistributionofSamplingdistributionofp1-

ofallvaluesIntervalEstimateIntervalEstimate

ofaPopulationProportionwhere:1-

istheconfidencecoefficient

z

/2isthezvalueprovidinganareaof

/2intheuppertailofthestandard normalprobabilitydistribution isthesampleproportionPoliticalScience,Inc.(PSI)specializesinvoterpollsandsurveysdesignedtokeeppoliticalofficeseekersinformedoftheirpositioninarace.Usingtelephonesurveys,PSIinterviewersaskregisteredvoterswhotheywouldvoteforiftheelectionwereheldthatday.IntervalEstimateofaPopulationProportionExample:PoliticalScience,Inc.Inacurrentelectioncampaign,PSIhasjustfoundthat220registeredvoters,outof500contacted,favoraparticularcandidate.PSIwantstodevelopa95%confidenceintervalestimatefortheproportionofthepopulationofregisteredvotersthatfavorthecandidate.IntervalEstimateofaPopulationProportionExample:PoliticalScience,Inc.where:n=500,=220/500=.44,z

/2=1.96IntervalEstimateofaPopulationProportionPSIis95%confidentthattheproportionofallvotersthatfavorthecandidateisbetween.3965and.4835.=.44+.0435Solvingforthenecessarysamplesize,wegetMarginofErrorSampleSizeforanIntervalEstimate

ofaPopulationProportionHowever,willnotbeknownuntilafterwehave