双样本的假设检验全英文版Two sample Tset of Hypothesis

TwoSampleTestsofHypothesisGOALSWhenyouhavecompletedthischapter,youwillbeableto:ONEUnderstandthedifferencebetweendependentandindependentsamples.THREEConductatestofhypothesisaboutthedifferencebetweentwoindependentpopulationmeanswhenatleastonesamplehaslessthan30observations.TWOConductatestofhypothesisaboutthedifferencebetweentwoindependentpopulationmeanswhenbothsampleshave30ormoreobservations.TwoSampleTestsofHypothesisGOALSWhenyouhavecompletedthischapter,youwillbeableto:FOUR

Conductatestofhypothesisregardingthedifferenceintwopopulationproportions.ComparingtwopopulationsWewishtoknowwhetherthedistributionofthedifferencesinsamplemeanshasameanof0.Ifbothsamplescontainatleast30observationsweusethezdistributionastheteststatistic.ComparingtwopopulationsNoassumptionsabouttheshapeofthepopulationsarerequired.Thesamplesarefromindependentpopulations.Theformulaforcomputingthevalueofzis:EXAMPLE1

Twocities,BradfordandKaneareseparatedonlybytheConewangoRiver.Thereiscompetitionbetweenthetwocities.ThelocalpaperrecentlyreportedthatthemeanhouseholdincomeinBradfordis$38,000withastandarddeviationof$6,000forasampleof40households.ThesamearticlereportedthemeanincomeinKaneis$35,000withastandarddeviationof$7,000forasampleof35households.Atthe.01significancelevelcanweconcludethemeanincomeinBradfordismore?EXAMPLE1continuedStep1:Statethenullandalternatehypotheses.

H0:µB≤µK;

H1:µB

>µKStep2:Statethelevelofsignificance.The.01significancelevelisstatedintheproblem.Step3:Findtheappropriateteststatistic.

Becausebothsamplesaremorethan30,wecanusezastheteststatistic.Example1continuedStep4:Statethedecisionrule.

Thenullhypothesisisrejectedifzisgreaterthan2.33.Step5:Computethevalueofzandmakeadecision.Example1continued

Thedecisionistonotrejectthenullhypothesis.WecannotconcludethatthemeanhouseholdincomeinBradfordislarger. Thep-valueis:

P(z>1.98)=.5000-.4761=.0239SmallSampleTestsofMeansThetdistributionisusedastheteststatisticifoneormoreofthesampleshavelessthan30observations.Therequiredassumptionsare: 1. Bothpopulationsmustfollowthenormaldistribution. 2. Thepopulationsmusthaveequalstandarddeviations. 3. Thesamplesarefromindependentpopulations.SmallsampletestofmeanscontinuedFindingthevalueoftheteststatisticrequirestwosteps. 1. Poolthesamplestandarddeviations.

Determinethevalueoftfromthefollowingformula.EXAMPLE2

ArecentEPAstudycomparedthehighwayfueleconomyofdomesticandimportedpassengercars.Asampleof15domesticcarsrevealedameanof33.7mpgwithastandarddeviationof2.4mpg.Asampleof12importedcarsrevealedameanof35.7mpgwithastandarddeviationof3.9.Atthe.05significancelevelcantheEPAconcludethatthempgishigherontheimportedcars?Example2continuedStep1:Statethenullandalternatehypotheses.

H0:µD

≥µI;

H1:µD

<µIStep2:Statethelevelofsignificance.The.05significancelevelisstatedintheproblem.Step3:Findtheappropriateteststatistic.

Bothsamplesarelessthan30,soweusethetdistribution.EXAMPLE2continued

Step5:Wecomputethepooledvariance:Step4:ThedecisionruleistorejectH0ift<-1.708.Thereare25degreesoffreedom.Example2continued

Wecomputethevalueoftasfollows.

H0isnotrejected.Thereisinsufficientsampleevidencetoclaimahighermpgontheimportedcars.TwoSampleTestsofProportionsWeinvestigatewhethertwosamplescamefrompopulationswithanequalproportionofsuccesses.Thetwosamplesarepooledusingthefollowingformula. whereX1andX2refertothenumberofsuccessesintherespectivesamplesofn1andn2. TwoSampleTestsofProportionscontinuedThevalueoftheteststatisticiscomputedfromthefollowingformula.Example4Areunmarriedworkersmorelikelytobeabsentfromworkthanmarriedworkers?Asampleof250marriedworkersshowed22missedmorethan5dayslastyear,whileasampleof300unmarriedworkersshowed35missedmorethanfivedays.Usea.05significancelevel.Example4continuedThenullandthealternatehypothesisare:H0:μU≤μM H1:μU>μM

Thenullhypothesisisrejectedifthecomputedvalueofzisgreaterthan1.65.Example4continuedThepooledproportionisThevalueoft