商业应用统计英文版ch17

1、AppliedBusinessStatistics,7thed.byKenBlack,Chapter17NonparametricStatistics,LearningObjectives,Recognizetheadvantagesanddisadvantagesofnonparametricstatistics.Understandhowtousetherunstesttotestforrandomness.KnowwhenandhowtousetheMann-WhitneyUtest,theWilcoxonmatched-pairssignedranktest,theKruskal-Wa

2、llistest,andtheFriedmantest.LearnwhenandhowtomeasurecorrelationusingSpearmansrankcorrelationmeasurement.,Parametricvs.NonparametricStatistics,ParametricStatisticsarestatisticaltechniquesbasedonassumptionsaboutthepopulationfromwhichthesampledataarecollected.Assumptionthatdatabeinganalyzedarerandomlys

3、electedfromanormallydistributedpopulation.Requiresquantitativemeasurementthatyieldintervalorratioleveldata.NonparametricStatisticsarebasedonfewerassumptionsaboutthepopulationandtheparameters.Sometimescalled“distribution-free”statistics.Avarietyofnonparametricstatisticsareavailableforusewithnominalor

4、ordinaldata.,AdvantagesofNonparametricTechniques,Sometimesthereisnoparametricalternativetotheuseofnonparametricstatistics.Certainnonparametrictestcanbeusedtoanalyzenominaldata.Certainnonparametrictestcanbeusedtoanalyzeordinaldata.Thecomputationsonnonparametricstatisticsareusuallylesscomplicatedthant

5、hoseforparametricstatistics,particularlyforsmallsamples.Probabilitystatementsobtainedfrommostnonparametrictestsareexactprobabilities.,DisadvantagesofNonparametricStatistics,Nonparametrictestscanbewastefulofdataifparametrictestsareavailableforusewiththedata.Nonparametrictestsareusuallynotaswidelyavai

6、lableandwellknowasparametrictests.Forlargesamples,thecalculationsformanynonparametricstatisticscanbetedious.,RunsTest,Testforrandomness-istheorderorsequenceofobservationsinasamplerandomornotEachsampleitempossessesoneoftwopossiblecharacteristicsRun-asuccessionofobservationswhichpossessthesamecharacte

7、risticExamplewithtworuns:F,F,F,F,F,F,F,F,M,M,M,M,M,M,MExamplewithfifteenruns:F,M,F,M,F,M,F,M,F,M,F,M,F,M,F,Samplesize:nNumberofsamplememberpossessingthefirstcharacteristic:n1Numberofsamplememberspossessingthesecondcharacteristic:n2n=n1+n2Ifbothn1andn2are20,thesmallsamplerunstestisappropriate.,RunsTe

8、st:SampleSizeConsideration,Suppose26coladrinkersaresampledrandomlytodeterminewhethertheypreferregularcolaordietcola.Therandomsamplecontains18regularcoladrinkersandeightdietcoladrinkers.LetCdenoteregularcoladrinkersandDdenotedietcoladrinkers.SupposethesequenceofsampledcoladrinkersisCCCCCDCCDCCCCDCDCC

9、CDDDCCC.Isthissequenceofcoladrinkersevidencethatthesampleisnotrandom?,RunsTest:SmallSampleExample,RunsTest:SmallSampleExample,H0:Theobservationsinthesamplearerandomlygenerated.Ha:Theobservationsinthesamplearenotrandomlygenerated.=.05n1=18n2=8If7R17,donotrejectH0Otherwise,rejectH0.123456789101112DCCC

10、CCDCCDCCCCDCDCCCDDDCCCR=12Since7R=1217,donotrejectH0,RunsTest:SmallSampleExample,Considerthefollowingmanufacturingexample.Amachineproducespartsthatareoccasionallyflawed.Whenthemachineisworkinginadjustment,flawsstilloccurbutseemtohappenrandomly.Aquality-controlpersonrandomlyselects50ofthepartsproduce

11、dbythemachinetodayandexaminesthemoneatatimeintheorderthattheyweremade.Theresultis40partswithnoflawsand10partswithflaws.Thesequenceofnoflaws(denotedbyN)andflaws(denotedbyF)isshownonanupcomingslide.Usinganalphaof.05,thequalitycontrollerteststodeterminewhetherthemachineisproducingrandomly(theflawsareoc

12、curringrandomly).,RunsTest:LargeSample,Ifeithern1orn2is20,thesamplingdistributionofRisapproximatelynormal.,RunsTest:LargeSample,RunsTest:LargeSampleExample,H0:Theobservationsinthesamplearerandomlygenerated.Ha:Theobservationsinthesamplearenotrandomlygenerated.=.05n1=40n2=10If-1.96Z1.96,donotrejectH0O

13、therwise,rejectH0.1123456789011NNNFNNNNNNNFNNFFNNNNNNFNNNNFNNNNN1213FFFFNNNNNNNNNNNNR=13,-1.96Z=-1.811.96,donotrejectH0,RunsTest:LargeSampleExample,RunsTest:LargeSampleExampleMINITABOutput,Mann-WhitneyUTest,Mann-WhitneyUtest-anonparametriccounterpartofthettestusedtocomparethemeansoftwoindependentpop

14、ulations.NonparametriccounterpartofthettestforindependentsamplesDoesnotrequirenormallydistributedpopulationsMaybeappliedtoordinaldataAssumptionsIndependentSamplesAtLeastOrdinalData,Sizeofsampleone:n1Sizeofsampletwo:n2Ifbothn1andn2are10,thesmallsampleprocedureisappropriate.Ifeithern1orn2isgreaterthan

15、10,thelargesampleprocedureisappropriate.,Mann-WhitneyUTest:SampleSizeConsideration,Isthereadifferencebetweenhealthserviceworkersandeducationalserviceworkersintheamountofcompensationemployerspaythemperhour?Supposearandomsampleofsevenhealthserviceworkersistakenalongwitharandomsampleofeighteducationals

16、erviceworkersfromdifferentpartsofthecountry.Eachoftheiremployersisinterviewedandfiguresareobtainedontheamountpaidperhourforemployeecompensationfortheseworkers.UseaMann-WhitneyUtesttodeterminewhetherthesetwopopulationsaredifferentinemployeecompensation.,Mann-WhitneyUTest:SmallSampleExample-Demonstrat

17、ionProblem17.1,H0:ThehealthservicepopulationisidenticaltotheeducationalservicepopulationonemployeecompensationHa:Thehealthservicepopulationisnotidenticaltotheeducationalservicepopulationonemployeecompensation,Mann-WhitneyUTest:SmallSampleExample-DemonstrationProblem17.1,=.05Ifthefinalp-value.05,reje

18、ctH0.W1=1+2+3+4+6+7+8=31W2=5+9+10+11+12+13+14+15=89,Mann-WhitneyUTest:SmallSampleExample-DemonstrationProblem17.1,Mann-WhitneyUTest:SmallSampleExample-DemonstrationProblem17.1MINITABOutput,SinceU2U1,U=3.p-value=.0011*2(foratwo-tailedtest)=.02215,Tisapproximatelynormallydistributed,andaZtestisused.If

19、n15,aspecial“smallsample”procedureisfollowed.Thepaireddataarerandomlyselected.Theunderlyingdistributionsaresymmetric.,WilcoxonMatched-PairsSignedRankTest:SampleSizeConsideration,ConsiderthesurveybyAmericanDemographicsthatestimatedtheaverageannualhouseholdspendingonhealthcare.TheU.S.metropolitanavera

20、gewas$1,800.SupposesixfamiliesinPittsburgh,Pennsylvania,arematcheddemographicallywithsixfamiliesinOakland,California,andtheiramountsofhouseholdspendingonhealthcareforlastyearareobtained.,WilcoxonMatched-PairsSignedRankTest:SmallSampleExample,H0:Md=0Ha:Md0n=6=0.05IfTobserved1,rejectH0.,WilcoxonMatche

21、d-PairsSignedRankTest:SmallSampleExample,WilcoxonMatched-PairsSignedRankTest:SmallSampleExample,NOTE:Differences=Pittsburg-Oakland,WilcoxonMatched-PairsSignedRankTest:MINITABOutput,Forlargesamples,theTstatisticisapproximatelynormallydistributedandazscorecanbeusedastheteststatistic.Thistechniquecanbe

22、appliedtotheairlineindustry,whereananalystmightwanttodeterminewhetherthereisadifferenceinthecostpermileofairfaresintheUnitedStatesbetween1979and2011forvariouscities.ThedatainTable17.3representthecostspermileofairlineticketsforasampleof17citiesforboth1979and2011.,WilcoxonMatched-PairsSignedRankTest:L

23、argeSampleFormulas,WilcoxonMatched-PairsSignedRankTest:LargeSampleFormulas,WilcoxonMatched-PairsSignedRankTest:LargeSampleFormulas,H0:Md=0Ha:Md0,AirlineCostDatafor17Cities,1979and2009,AirlineCost:TCalculation,AirlineCost:Conclusion,Kruskal-WallisTest-Anonparametricalternativetoone-wayanalysisofvaria

24、nceMayusedtoanalyzeordinaldataNoassumedpopulationshapeAssumesthattheCgroupsareindependentAssumesrandomselectionofindividualitems,Kruskal-WallisTest,Kruskal-WallisKStatistic,NumberofPatientsperDayperPhysicianinThreeOrganizationalCategories,Supposearesearcherwantstodeterminewhetherthenumberofphysician

25、sinanofficeproducessignificantdifferencesinthenumberofofficepatientsseenbyeachphysicianperday.Shetakesarandomsampleofphysiciansfrompracticesinwhich(1)thereareonlytwopartners,(2)therearethreeormorepartners,or(3)theofficeisahealthmaintenanceorganization(HMO).,Ho:ThethreepopulationsareidenticalHa:Atlea

26、stoneofthethreepopulationsisdifferent,NumberofPatientsperDayperPhysicianinThreeOrganizationalCategories,PatientsperDayData:Kruskal-WallisPreliminaryCalculations,PatientsperDayData:Kruskal-WallisCalculationsandConclusion,PhysiciansPatientsData:Kruskal-WallisMINITABOutput,FriedmanTest,FriedmanTest-Ano

27、nparametricalternativetotherandomizedblockdesignAssumptionsTheblocksareindependent.Thereisnointeractionbetweenblocksandtreatments.Observationswithineachblockcanberanked.HypothesesHo:ThetreatmentpopulationsareequalHa:Atleastonetreatmentpopulationyieldslargervaluesthanatleastoneothertreatmentpopulatio

28、n,FriedmanTest,Amanufacturingcompanyassemblesmicrocircuitsthatcontainaplastichousing.Managersareconcernedaboutanunacceptablyhighnumberoftheproductsthatsustainedhousingdamageduringshipment.Thehousingcomponentismadebyfourdifferentsuppliers.Managershavedecidedtoconductastudyoftheplastichousingbyrandoml

29、yselectingfivehousingsmadebyeachofthefoursuppliers.Onehousingisselectedforeachdayoftheweek.Thatis,foreachsupplier,ahousingmadeonMondayisselected,onemadeonTuesdayisselected,andsoon.Inanalyzingthedata,thetreatmentvariableissupplierandthetreatmentlevelsarethefoursuppliers.Theblockingeffectisdayofthewee

30、kwitheachdayrepresentingablocklevel.Thequalitycontrolteamwantstodeterminewhetherthereisanysignificantdifferenceinthetensilestrengthoftheplastichousingbysupplier.,FriedmanTest:TensileStrengthofPlasticHousings,Ho:ThesupplierpopulationsareequalHa:Atleastonesupplierpopulationyieldslargervaluesthanatleastoneothersupplierpopulation,FriedmanTest:TensileStrengthofPlasticHousings,FriedmanTest:TensileStrengthofPlasticHousings,FriedmanTest:TensileStrengthofP