BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-1Chapter10Two-SampleTestsBusinessStatistics:AFirstCourse

FifthEditionBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-2LearningObjectivesInthischapter,youlearn:

HowtousehypothesistestingforcomparingthedifferencebetweenThemeansoftwoindependentpopulationsThemeansoftworelatedpopulationsTheproportionsoftwoindependentpopulationsThevariancesoftwoindependentpopulationsBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-3Two-SampleTestsTwo-SampleTestsPopulationMeans,IndependentSamplesPopulationMeans,RelatedSamplesPopulationVariancesMean1vs.Mean2Samegroupbeforevs.aftertreatmentVariance1vs.Variance2Examples:PopulationProportionsProportion1vs.Proportion2BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-4DifferenceBetweenTwoMeansPopulationmeans,independentsamplesGoal:Testhypothesisorformaconfidenceintervalforthedifferencebetweentwopopulationmeans,μ1–μ2

ThepointestimateforthedifferenceisX1–X2*σ1andσ2unknown,assumedequalσ1andσ2unknown,notassumedequalBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-5DifferenceBetweenTwoMeans:IndependentSamplesPopulationmeans,independentsamples*UseSptoestimateunknownσ.UseaPooled-Variance

ttest.σ1andσ2unknown,assumedequalσ1andσ2unknown,notassumedequalUseS1andS2toestimateunknownσ1andσ2.UseaSeparate-variancettestDifferentdatasourcesUnrelatedIndependentSampleselectedfromonepopulationhasnoeffectonthesampleselectedfromtheotherpopulationBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-6HypothesisTestsfor

TwoPopulationMeansLower-tailtest:H0:μ1

μ2H1:μ1<μ2i.e.,H0:μ1–μ2

0H1:μ1–μ2

<0Upper-tailtest:H0:μ1≤μ2H1:μ1

>

μ2i.e.,H0:μ1–μ2

≤0H1:μ1–μ2

>0Two-tailtest:H0:μ1=μ2H1:μ1

≠

μ2i.e.,H0:μ1–μ2

=0H1:μ1–μ2

≠0TwoPopulationMeans,IndependentSamplesBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-7TwoPopulationMeans,IndependentSamplesLower-tailtest:H0:μ1–μ2

0H1:μ1–μ2

<0Upper-tailtest:H0:μ1–μ2

≤0H1:μ1–μ2

>0Two-tailtest:H0:μ1–μ2

=0H1:μ1–μ2

≠0aa/2a/2a-ta-ta/2tata/2RejectH0iftSTAT<-taRejectH0iftSTAT>taRejectH0iftSTAT<-ta/2

ortSTAT>ta/2

Hypothesistestsforμ1–μ2

BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-8Populationmeans,independentsamplesHypothesistestsforμ1-μ2withσ1andσ2unknownandassumedequalAssumptions:

SamplesarerandomlyandindependentlydrawnPopulationsarenormallydistributedorbothsamplesizesareatleast30Populationvariancesareunknownbutassumedequal*σ1andσ2unknown,assumedequalσ1andσ2unknown,notassumedequalBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-9Populationmeans,independentsamplesThepooledvarianceis:Theteststatisticis:WheretSTAThasd.f.=(n1+n2–2)(continued)*σ1andσ2unknown,assumedequalσ1andσ2unknown,notassumedequalHypothesistestsforμ1-μ2withσ1andσ2unknownandassumedequalTheconfidenceintervalfor

μ1–μ2is:Wheretα/2hasd.f.=n1+n2–2BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-10Populationmeans,independentsamples*Confidenceintervalforμ1-μ2withσ1andσ2unknownandassumedequalσ1andσ2unknown,assumedequalσ1andσ2unknown,notassumedequalBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-11Pooled-VariancetTestExampleYouareafinancialanalystforabrokeragefirm.IsthereadifferenceindividendyieldbetweenstockslistedontheNYSE&NASDAQ?Youcollectthefollowingdata:

NYSE

NASDAQ

Number2125Samplemean 3.272.53Samplestddev 1.301.16Assumingbothpopulationsareapproximatelynormalwithequalvariances,is

thereadifferenceinmean

yield(

=0.05)?BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-12Pooled-VariancetTestExample:CalculatingtheTestStatisticTheteststatisticis:(continued)H0:μ1-μ2=0i.e.(μ1=μ2)H1:μ1-μ2≠0i.e.(μ1≠μ2)BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-13Pooled-VariancetTestExample:HypothesisTestSolutionH0:μ1-μ2=0i.e.(μ1=μ2)H1:μ1-μ2≠0i.e.(μ1≠μ2)

=0.05df=21+25-2=44CriticalValues:t=±2.0154TestStatistic:Decision:Conclusion:RejectH0ata=0.05Thereisevidenceofadifferenceinmeans.t02.0154-2.0154.025RejectH0RejectH0.0252.040BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-14Pooled-VariancetTestExample:ConfidenceIntervalforμ1-μ2SincewerejectedH0canwebe95%confidentthatμN(yùn)YSE>μN(yùn)ASDAQ?95%ConfidenceIntervalforμN(yùn)YSE-μN(yùn)ASDAQSince0islessthantheentireinterval,wecanbe95%confidentthatμN(yùn)YSE>μN(yùn)ASDAQ

OwensLawnCare,Inc.,manufacturesandassembleslawnmowersthatareshippedtodealersthroughouttheUnitedStatesandCanada.Twodifferentprocedureshavebeenproposedformountingtheengineontheframeofthelawnmower.Thequestionis:Isthereadifferenceinthemeantimetomounttheenginesontheframesofthelawnmowers?ThefirstprocedurewasdevelopedbylongtimeOwensemployeeHerbWelles(designatedasprocedure1),andtheotherprocedurewasdevelopedbyOwensVicePresidentofEngineeringWilliamAtkins(designatedasprocedure2).Toevaluatethetwomethods,itwasdecidedtoconductatimeandmotionstudy.ComparingPopulationMeanswith

UnknownPopulationStandardDeviations

(thePooledt-test)-ExampleBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-15AsampleoffiveemployeeswastimedusingtheWellesmethodandsixusingtheAtkinsmethod.Theresults,inminutes,areshownontheright.Isthereadifferenceinthemeanmountingtimes?Usethe.10significancelevel.Step1:Statethenullandalternatehypotheses.

H0:μ1=μ2

H1:μ1

≠μ2thePooledt-testExampleBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-16Step2:Statethelevelofsignificance.The.10significancelevelisstatedintheproblem.Step3:Findtheappropriateteststatistic.

Assumingbothpopulationsareapproximatelynormalwithequalvariances,weusethepooledt-test.(continued)Step4:Statethedecisionrule.

RejectH0ift>t/2,n1+n2-2ort<-t/2,n1+n2-2 t>t.05,9ort<-t.05,9 t>1.833

ort<-1.833

BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-17thePooledt-testExample(continued)Step5:Computethevalueoftandmakeadecision(a)CalculatethesamplestandarddeviationsBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-18thePooledt-testExample(continued)Step5:Computethevalueoftandmakeadecision-0.662Thedecisionisnottorejectthenullhypothesis,because0.662fallsintheregionbetween-1.833and1.833.Weconcludethatthereisnodifferenceinthemeantimestomounttheengineontheframeusingthetwomethods.BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-19thePooledt-testExample(continued)BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-20thePooledt-testExampleArecentstudy(“SnackAdsSpurChildrentoEatMore,”TheNewYorkTimes,July20,2009,p.B3)foundthatchildrenwhowatchedacartoonwithfoodadvertisingate,onaverage,28.5gramsofGoldfishcrackersascomparedtoanaverageof19.7gramsofGoldfishcrackersforchildrenwhowatchedacartoonwithoutfoodadvertising.Supposethattherewere59childrenineachgroup,andthesamplestandarddeviationforthosechildrenwhowatchedthefoodadwas8.6gramsandthesamplestandarddeviationforthosechildrenwhodidnotwatchthefoodadwas7.9grams.BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-21thePooledt-testExample(continued)a.Assumingthatthepopulationvariancesareequalandα=0.05,isthereevidencethatthemeanamountofGoldfishcrackerseatenwassignificantlyhigherforthechildrenwhowatchedfoodads?b.Assumingthatthepopulationvariancesareequal,constructa95%confidenceintervalestimateofthedifferencebetweenthemeanamountofGoldfishcrackerseatenbythechildrenwhowatchedanddidnotwatchthefoodad.BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-22thePooledt-testExample(continued)a.Becausethepopulationstandarddeviationsarenotknownbutareassumedtobeequal,weusethepooledt-test.RejectH0ifSoourdecisionistorejectH0.>1.6588BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-23thePooledt-testExample(continued)Let95%ConfidenceIntervalforμ1-μ2b.BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-24Populationmeans,independentsamplesHypothesistestsforμ1-μ2withσ1andσ2unknown,notassumedequalAssumptions:

SamplesarerandomlyandindependentlydrawnPopulationsarenormallydistributedorbothsamplesizesareatleast30Populationvariancesareunknownandcannotbeassumedtobeequal*σ1andσ2unknown,assumedequalσ1andσ2unknown,notassumedequalBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-25Populationmeans,independentsamples(continued)*σ1andσ2unknown,assumedequalσ1andσ2unknown,notassumedequalHypothesistestsforμ1-μ2withσ1andσ2unknownandnotassumedequalTheteststatisticforμ1–μ2is:BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-26RelatedPopulations

ThePairedDifferenceTest

TestsMeansof2RelatedPopulations PairedormatchedsamplesRepeatedmeasures(before/after)Usedifferencebetweenpairedvalues:EliminatesVariationAmongSubjectsAssumptions:BothPopulationsAreNormallyDistributedOr,ifnotNormal,uselargesamplesRelatedsamplesDi=X1i-X2iBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-27RelatedPopulations

ThePairedDifferenceTestTheithpaireddifferenceisDi,whereRelatedsamplesDi=X1i-X2i

ThepointestimateforthepaireddifferencepopulationmeanμDisD:nisthenumberofpairsinthepairedsampleThesamplestandarddeviationisSD(continued)BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-28TheteststatisticforμDis:PairedsamplesWheretSTAThasn-1d.f.ThePairedDifferenceTest:

FindingtSTATBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-29Lower-tailtest:H0:μD

0H1:μD<0Upper-tailtest:H0:μD≤0H1:μD

>0Two-tailtest:H0:μD=0H1:μD

≠0PairedSamplesThePairedDifferenceTest:PossibleHypothesesaa/2a/2a-ta-ta/2tata/2RejectH0iftSTAT<-taRejectH0iftSTAT>taRejectH0iftSTAT<-ta/2

ortSTAT>ta/2

WheretSTAThasn-1d.f.BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-30TheconfidenceintervalforμDisPairedsampleswhereThePairedDifferenceConfidenceIntervalBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-31Assumeyousendyoursalespeopletoa“customerservice”trainingworkshop.Hasthetrainingmadeadifferenceinthenumberofcomplaints?Youcollectthefollowingdata:PairedDifferenceTest:Example

NumberofComplaints:

(2)-(1)Salesperson

Before(1)

After(2)

Difference,

DiC.B. 6

4-2T.F. 20

6-14M.H. 3

2-1R.K. 0

00M.O. 4

0

-4 -21D=

Din

=-4.2BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-32Hasthetrainingmadeadifferenceinthenumberofcomplaints(atthe0.01level)?

-4.2D=H0:μD=0H1:

μD

0TestStatistic:t0.005=±4.604

d.f.=n-1=4Reject

/2

-4.6044.604Decision:

DonotrejectH0(tstatisnotintherejectregion)Conclusion:

Thereisnotasignificantchangeinthenumberofcomplaints.PairedDifferenceTest:Solution

Reject

/2

-1.66

=.01NickelSavingsandLoanwishestocomparethetwocompaniesitusestoappraisethevalueofresidentialhomes.NickelSavingsselectedasampleof10residentialpropertiesandscheduledbothfirmsforanappraisal.Theresults,reportedin$000,areshownonthetable(right).Atthe.05significancelevel,canweconcludethereisadifferenceinthemeanappraisedvaluesofthehomes?BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-33HypothesisTestingInvolving

PairedObservations-ExampleStep1:Statethenullandalternatehypotheses.

H0:

d=0

H1:

d

≠

0Step2:Statethelevelofsignificance. The.05significancelevelisstatedintheproblem.Step3:Findtheappropriateteststatistic.

Assumingbothpopulationsareapproximatelynormal,Wewillusethet-test.HypothesisTestingInvolving

PairedObservations-ExampleBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-34(continued)Step4:Statethedecisionrule.

RejectH0if

t>t/2,n-1ort<-t/2,n-1

t>t.025,9ort<-t.025,9

t>2.262ort<-2.262HypothesisTestingInvolving

PairedObservations-Example(continued)BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-35Step5:Computethevalueoftandmakeadecision

Thecomputedvalueoftisgreaterthanthehighercriticalvalue,soourdecisionistorejectthenullhypothesis.Weconcludethatthereisadifferenceinthemeanappraisedvaluesofthehomes.HypothesisTestingInvolving

PairedObservations-Example(continued)BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-36NineexpertsratedtwobrandsofColombiancoffeeinataste-testingexperiment.AratingonaProblemsfor7-pointscale(1=extremelyunpleasing,7=extremelypleasing)isgivenforeachoffourcharacteristics:taste,aroma,richness,andacidity.ThefollowingdatastoredinCoffeecontaintheratingsaccumulatedoverallfourcharacteristics:BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-37HypothesisTestingInvolving

PairedObservations-ExampleExpertBrandABrandBC.C.2426S.E.2727E.G.1922B.L.2427C.M.2225C.N.2627G.N.2726R.M.2527P.V.2223BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-38HypothesisTestingInvolving

PairedObservations-Example(continued)b.Atthe0.05levelofsignificance,isthereevidenceofadifferenceinthemeanratingsbetweenthetwobrands?a.Whatassumptionisnecessaryaboutthepopulationdistributioninordertoperformthetestof(b)?c.Constructandinterpreta95%confidenceintervalestimateofthedifferenceinthemeanratingsbetweenthetwobrands.BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-39HypothesisTestingInvolving

PairedObservations-Example(continued)Assumingbothpopulationsareapproximatelynormal,Wewillusethet-testb.RejectH0ifBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-40HypothesisTestingInvolving

PairedObservations-Example(continued)ExpertBrandABrandBDC.C.242620.44440.1975S.E.27270-1.55562.4199E.G.192231.44442.0863B.L.242731.44442.0863C.M.222531.44442.0863C.N.26271-0.55560.3087G.N.2726-1-2.55566.5311R.M.252720.44440.1975P.V.22231-0.55560.3087BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-41HypothesisTestingInvolving

PairedObservations-Example(continued)SoourdecisionistorejectH0.95%ConfidenceIntervalforμDc.BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-42TwoPopulationProportionsGoal:testahypothesisorformaconfidenceintervalforthedifferencebetweentwopopulationproportions, π1–π2

ThepointestimateforthedifferenceisPopulationproportionsAssumptions:

n1π1

5,n1(1-π1)5n2π2

5,n2(1-π2)5

BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-43TwoPopulationProportionsPopulationproportionsThepooledestimatefortheoverallproportionis:whereX1andX2arethenumberofitemsofinterestinsamples1and2Inthenullhypothesisweassumethenullhypothesisistrue,soweassumeπ1=π2andpoolthetwosampleestimatesBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-44TwoPopulationProportionsPopulationproportionsTheteststatisticforπ1–π2isaZstatistic:(continued)whereBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-45HypothesisTestsfor

TwoPopulationProportionsPopulationproportionsLower-tailtest:H0:π1

π2H1:π1<π2i.e.,H0:π1–π2

0H1:π1–π2

<0Upper-tailtest:H0:π1≤π2H1:π1

>

π2i.e.,H0:π1–π2

≤0H1:π1–π2

>0Two-tailtest:H0:π1=π2H1:π1

≠

π2i.e.,H0:π1–π2

=0H1:π1–π2

≠0BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-46HypothesisTestsfor

TwoPopulationProportionsPopulationproportionsLower-tailtest:H0:π1–π2

0H1:π1–π2

<0Upper-tailtest:H0:π1–π2

≤0H1:π1–π2

>0Two-tailtest:H0:π1–π2

=0H1:π1–π2

≠0aa/2a/2a-za-za/2zaza/2RejectH0ifZSTAT<-ZaRejectH0ifZSTAT>ZaRejectH0ifZSTAT<-Za/2

orZSTAT>Za/2

(continued)BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-47ConfidenceIntervalfor

TwoPopulationProportionsPopulationproportionsTheconfidenceintervalfor

π1–π2is:BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-48HypothesisTestExample:

TwopopulationProportionsIsthereasignificantdifferencebetweentheproportionofmenandtheproportionofwomenwhowillvoteYesonPropositionA?Inarandomsample,36of72menand31of50womenindicatedtheywouldvoteYesTestatthe.05levelofsignificanceBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-49Thehypothesistestis:H0:π1–π2

=0(thetwoproportionsareequal)H1:π1–π2

≠0(thereisasignificantdifferencebetweenproportions)Thesampleproportionsare:Men: p1=36/72=.50Women: p2=31/50=.62Thepooledestimatefortheoverallproportionis:HypothesisTestExample:

TwopopulationProportions(continued)BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-50Theteststatisticforπ1–π2is:HypothesisTestExample:

TwopopulationProportions(continued).025-1.961.96.025-1.31Decision:

DonotrejectH0Conclusion:

Thereisnotsignificantevidenceofadifferenceinproportionswhowillvoteyesbetweenmenandwomen.RejectH0RejectH0CriticalValues=±1.96For=.05BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-51HypothesisTestExample:

TwopopulationProportionsHowdoAmericansfeelaboutadsonwebsites?Asurveyof1,000adultInternetusersfoundthat670opposedadsonwebsites.(DataextractedfromS.Clifford,“TackedforAds?ManyAmericansSayNoThanks,”TheNewYorkTimes,September30,2009,p.B3.)Supposethatasurveyof1,000Internetusersage12–17foundthat510opposedadsonwebsites.a.Atthe0.05levelofsignificance,isthereevidenceofadifferencebetweenadultInternetusersandInternetusersage12–17intheproportionwhoopposeads?b.Findthep-valuein(a).BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-52HypothesisTestingInvolving

PairedObservations-Example(continued)WewillusetheZ-testa.RejectH0ifBusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-53HypothesisTestingInvolving

PairedObservations-Example(continued)SoourdecisionistorejectH0.b.BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-54HypothesisTestsforVariancesTestsforTwoPopulationVariancesFteststatisticH0:σ12=σ22H1:σ12≠σ22H0:σ12≤σ22H1:σ12>σ22*Hypotheses FSTATS12/S22S12=Varianceofsample1(thelargersamplevariance)n1=samplesizeofsample1S22=Varianceofsample2(thesmallersamplevariance)n2=samplesizeofsample2n1–1=numeratordegreesoffreedomn2–1=denominatordegreesoffreedomWhere:BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-55TheFcriticalvalue

isfoundfromtheFtableTherearetwodegreesoffreedomrequired:numeratoranddenominatorWhenIntheFtable,numeratordegreesoffreedomdeterminethecolumndenominatordegreesoffreedomdeterminetherowTheFDistributiondf1=n1–1;df2=n2–1BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-56FindingtheRejectionRegionH0:σ12=σ22H1:σ12≠σ22H0:σ12≤σ22H1:σ12>σ22F

0

Fα

RejectH0DonotrejectH0RejectH0ifFSTAT>FαF

0

/2RejectH0DonotrejectH0Fα/2

RejectH0ifFSTAT>Fα/2BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-57FTest:AnExampleYouareafinancialanalystforabrokeragefirm.YouwanttocomparedividendyieldsbetweenstockslistedontheNYSE&NASDAQ.Youcollectthefollowingdata:

NYSE

NASDAQ

Number 21 25Mean 3.27 2.53Stddev 1.30 1.16Isthereadifferenceinthe variancesbetweentheNYSE &NASDAQatthe

=

0.05level?BusinessStatistics:AFirstCourse,5e?2009Prentice-Hall,Inc.Chap10-58FTest:ExampleSolutionFormthehypothesistest:H0:σ21=σ22(thereisnodifferencebetweenvariances)H1:σ21≠σ22(thereisadifferencebetweenvariances)FindtheFcriticalvaluefor

=0.05:Numeratord.f.=n1–1=21–1=20Denomin