统计专业英语 Chapter-8-Hypothesis-Testing

statisticsinpracticeWithourcountrytakingpartinWTO，ourcompaniesarefacingextraordinarilyseriouschallenges，particularlyinAutomobileIndustry。Itisnotonlychallenge，butalsotheopportunity。Inordertoacceptthechallenges，nationalAutomobileIndustry，insuccession，takelotsofmeasurestoreplyit。AautogroupcarriedthroughaseriesofimprovementsonenginesystemonA1modelcar。Eventually，itincreasesthespeedofstartupandknocksdownthenoises，renamedbyA2。Duringthis，onemomentousissuethatcompanypaidattentiontoisenergyefficientofcars。Savinggasisbigsalepoint。TheA1carthatunimprovedcostrelativelyhighvolumegas，8。48literperonehundredkilometers。Thecompanywishthatimprovedcarscansavethegasthantheformercars，atleast，notcostmorethanbefore。Chapter8HypothesisTestingChapter8StatisticsinpracticeTherefore，takeout15A2autosrandomlytodoexperiment，getthestatisticsthat15carscostthevolumeofgasper100kilometers。Seethefollowingchart：（15autosconsumethevolumeofgasper100kilometers）unit：liter8.508.758.338.218.528.308.318.198.408.868.418.018.208.268.39Theaverageis8.377.accordingtothisstatistics,themanagerofTechnologyApartmentconfirmthatimprovedautoscansavethegas.ThemanagerofQualityApartmenthasdifferentideaaboutthisconclusion.Hethoughtthatthephenomenoncomefromtherandomsample.Itissoearlytoperoratethatimprovedautosincreasethemeanmiles-per-gallonratingandweshoulddoHypothesisTestingonstatistics.EngineerZhangwhohaspassedthepractitionerqualificationfromtheQualityApartmenthaslearntalotofstatisticsmethods.ThemanagerofQualityApartmentaskedEngineerZhangtosolvethisproblem.Throughsimplecalculation,theEngineerZhanggottheconclusionthatthepresentdatacannotfiguretheautosbeforeandafterimprovedhavenodistinctchangesinmiles-per-gallonrating.Therefore,howEngineerZhangtogetthisconclusion?Chapter8StatisticsinpracticeThekeystoneofchapter1、thebasictheoryofHypothesisTesting2、theformandkindofHypothesisTesting；3、typeIandtypeIIerror；4、IntervalTestingandmethodsofHypothesisTesting。Thedifficultiesofchapter1、thebasictheoryofHypothesisTesting；2、typeIandtypeIIerrorChapter8Statisticsinpractice8.1.1TestingresearchHypothesisTheformercasecanbeseenanexampleofTestingResearchhypothesesResearchhypothesis:thenewcarswillprovideameanmiles-per-gallonbelower,thatis,theaverageofcostinggasislower8.48literIngeneral，researchhypothesescanbeseenalternativeHypothesis，SowecansetupfollowingnullandalternativeHypothesisChapter8DevelopingnullandalternativeHypothesisExample:amanufacturerofsoftdrinkswhostatesthattwo-litercontainersofitsproductshaveanaverageofatleast67.6fluidounces.Asampleoftwo-litercontainerswillbeselected,andthecontentswillbemeasuredtotestthemanufacture'claim.Inthistypeofhypothesis-testingsituation,wegenerallybeginbyassumingthatthamanufacturer’sclaimistrue.Usingthisapproach,forthesoft-drinkexample,wewouldstatethenullandalternativehypothesesasfollows.Chapter8testingthevalidityofaclaim

8.1.3testinginDecision-MakingsituationsIngeneral,adecision-makermustchoosebetweennullandalternativehypothesis.Example:onthebasisofasampleofpartsfromshipmentthathasjustbeenreceived,aquality-controlinspectormustdecidewhethertoaccepttheentireshipmentortoreturntheshipmenttothesupplierbecauseitdoesnotmeetspecification.Assumethatspecificationforaparticularpartstatethatameanlengthoftwoinchesisdesired.Inthiscase,thenullandalternativehypotheseswouldbeformulatedasfollows.

Letdenotethespecificnumericalvaluebeingconsideredinthenullandalternativehypothesis.Ingeneral,ahypothesistestaboutthevaluesofapopulationmusttakeoneofthefollowingthreeforms:LefttailedtesttwotailedtestRighttailedtestSummaryofFormsfornullandAlternativeHypothesesChapter8TypeIandTypeTTerrorsTypeI：rejectthecorrectoriginalHypothesis，calledProducer'sRisk

TypeII：acceptthewrongoriginalHypothesis，calledConsumer’sRisk

PopulationconditionconclusionHotrueHatrueAcceptH0correcttypeTTerrorconclusionRejectH0TypeIerrorcorrectconclusionWedenotetheprobabilitiesofmakingthetwoerrorsasfollows:α——theprobabilityofmakingaTypeIerrorβ——theprobabilityofmakingaTypeTTerrorInpractice，thepersonconductingthehypothesistestspecifiesthemaximumallowableprobabilityofmakingaTypeIerror，calledthelevelofsignificanceforthetest。Commonchoicesforthelevelofsignificanceareα=0.05orα=0.01。Chapter8TypeIandTypeTTerrorsOne—tailedtestsaboutapopulationmean：Large—samplecaseExample：theFederalTradecommissionperiodicallyconductsstudiesdesignedtotesttheclaimsmanufacturersmakeabouttheirproducts.Forexample,thelabelonalargecanofHilltopcoffeestatesthatthecancontainsatleastthreepoundsofcoffee.Supposewewanttocheckthisclaimbyusinghypothesistesting.supposearandomsampleof36cansofcoffeeisselected.Steps:Thefirststepistodevelopthenullandalternativehypothesis.Notethatifthemeanfillingweightforthesampleof36cansislessthanthreepounds.thesampleresultswillbegintocastaboutonthenullhypothesis.ButhowmuchlessthanthreepoundsmustlessthanthreepoundsmustXbebeforewewouldbewillingtoriskmakingaTypeIerrorandfalselyaccusethecompanyofalabelviolation.ItalldependsontheattitudeofmanagersFirstweconsidertheconditionof,thefigurefollowedshowstheprobabilitythatsamplemeanislessthanthestandarddeviationof1.645timesofpopulationmeanis0.05.IftheFTCconsiderthatthe0.05probabilityofhappeningthetypeIerrorisacceptable,aslongasthevalueofstatistic-----zshowsthatthesamplemeanislessthanthestandarddeviationof1.645timesofpopulationmean,wecanrejectthenullhypotheses,thatisWhenn=36,thesamplemeanisinnormaldistribution,sowecanusethevalueofstatistictomeasurethedepartureofsamplemeanfrompopulationmean.rejectOne—tailedtestsaboutapopulationmean：

Large—samplecaseTheprobabilitythatsampleaverageismorethat1.645standarddeviationsThesampledistributionofOne—tailedtestsaboutapopulationmean：Large—samplecaseBeforegoingtesting,weshouldspecifythemaximumallowableprobabilityofatypeIerror,calledthelevelsignificanceforthetest.Inthehilltopcoffeestudy,thedirectoroftheFTC’stestingprogramhasmadethefollowingstatement:ifthecompanyismeetingitsweightspecificationsexactlyWewouldlikea99%chanceofnottakinganyactionagainstthecompany.whilewedonotwanttoaccusethecompanywronglyofunderfillingitsproducts,wearewillingtoriska1%chanceofmakingthiserror.Wecanconclude,Fromthenormaldistributiontable,wecangetthecriticalvalue2.33。One—tailedtestsaboutapopulationmean：Large—samplecaseAccordingtothesampleaverage,wecangetthattheZislessthan-2.33.thenwecanrejectthenullhypothesisandacceptthealternativehypothesis.Supposeasampleof36cansprovideameanofPoundsandweknowfrompreviousstudiesthatthepopulationdeviation

Z<-2.33rejectedareaIt’sintherejectionregion,sowecanrejectnullhypotheses.One—tailedtestsaboutapopulationmean：Large—samplecaseif，thenthevalueofstatistic

if，theprobabilityoftypeIerrorislessthanthetypeIIerror.Theprobabilitythatthevalueofteststatisticisintherejectionregionisevensmaller.so,togetthecriticalvalueoftest,weonlyneedtoassumeIt’sintherejectionregion,sowecannotrejectnullhypotheses.One—tailedtestsaboutapopulationmean：Large—samplecaseChapter8One-tailedtestsaboutapopulationmeansummary:inlargecase,whetherteststatisticisknownornot,thesampleaverageobeythenominaldistributionThegeneralformofalower-taillefttest.1)setupnullandalternativehypothesis:2)Ensuretheteststatisticandcalculateit:3)Accordingtothepreviouslevelofsignificance,checkthenominaldistributioncharttofindcriticalvalueRejectionruleatalevelofsignificance

orreject

Asthesame,inlargecase,Thegeneralformofalower-tailrighttestis:1)setupnullandalternativehypothesis:2)Ensuretheteststatisticandcalculateit:3)Accordingtothepreviouslevelofsignificance,checkthenominaldistributioncharttofindcriticalvalue.4)RejectionruleatalevelofsignificancerejectorChapter8One-tailedtestsaboutapopulationmeanCase:Onecompanyproducedanewtypeoftire.Itsdesignstandardwastohavetheabilitytorunatleast28000kilometersperhour.30tireswerepickedoutaccidentallyassamplesforexperiment,asaresult,thesamplemeanwas27500kilometersperhour,andthesamplestandarddeviationwas1000kilometersperhour.thelevelofsignificanceis0.05,testifthereisenoughprooftobeagainsttherepresentthattheaveragedistanceperhourisatleast28000kilometers.Solution:weknowthat1,setthenullandalternativehypothesesChapter8One-tailedtestsaboutapopulationmean2.Gettheteststatistic,andcalculateitsvalue3.4.Rejectsowecannotacceptthecompany’srepresentaboutthetireExercise:P272,T14Chapter8One-tailedtestsaboutapopulationmeanChapter8TheUseofP-valueThevalueofPistheprobabilitythatthesamplemeanislessthanorequaltotheobservedvalue.Anditcanalsobecalledtheobservedlevelofsignificance.SetthecaseofHillCo.’scoffeeproblemforexampletocalculateP-valueofthesamplemeanWehavetheteststatisticz=-2.67,andintheformofnormaldistribution,wecangettheareabetweenmeanandz=-2.67is0.4962.sowegettheprobabilitythatthesamplemeanislessthanorequaltotheobservedvalueis0.5000-0.4962=0.0038,andP-valueis0.0038.P-valuecanbeusedtomakedecisionofHypothesisTesting,ifP-valueislessthanlevelofsignificance,thevalueofteststatisticwillbeinrejectionregion,ifit’smorethanlevelofsignificance,thevalueofteststatisticwillbeinacceptedregion.Intheexampleabove,P=0.0038,nullhypothesesshouldberejectedTheP-valuestandardoftesthypothesesnullhypothesesshouldberejectedChapter8TheUseofP-valueChapter8Two-TailedTestsinlarge-SamplecaseCase:AstheruleofAmericanGolfClub,onlythegolfballwhoseaveragevalueofrangeandtherollingdistanceis280meterscanbeusedinthegame.Assumeacompanydevelopedaadvancedproducingtechnology,whichcanmakegolfballswhoseaveragevalueofrangeandtherollingdistanceare280meters.Nowwepickout36golfballsasrandomsamplestotestifthecompany’srepresentistrue.Thedataisasfollowed.(Supposeit’sproceededontheobservedlevelofsignificancewhichis0.05.269301296275282276284272263300295265282263286260285264268288271260270293299293273278278279266269274277281291Thisisanexampleoftwo-tailedtests.First,setupthenullandalternativehypothesisasfollows:Inlargecase,westillusestatisticvalueZWhat’sdifferenttoone-tailedtestistherejectionregionisdistributedonbothsidesofnormaldistributioncurve,thecorrespondingprobabilitybothare.Whenyoulookuptheform,youshouldfindout’scorrespondingcriticalvalueChapter8Two-TailedTestsinlarge-SamplecaseIntheexampleabove,wecangetfromthedataintheform,

SothevalueofstatisticisAsthegivenlevelofobservationInthetable,It’sintheacceptedregion,andwecannotrejectnullhypothesesChapter8Two-TailedTestsinlarge-SamplecaseIntheexampleabove,wecangetfromthedataintheform,1,setupnullandreservationhypotheses2,Gettheteststatistic,andcalculateitsvalue3,asthelevelofobservation,lookupthecriticalvaluefromthenormaldistributiontable.4,theruleofrejection:ororrejectChapter8Two-TailedTestsinlarge-SamplecaseAbouttheexampleofgolfabove,wehaveknownthecorrespondingsamplemean’sZ-valueis-0.75，fromthenormaldistributiontable,theareabetweenmeanandz-value-0.75is0.2734.Asaresult,theleftareais0.2266,andtheareaofleftrejectionregionis=0.025.0.2266>0.025，thestatisticisnotintherejectionregion,sowecannotrejectnullhypotheses,anditisnotagainstthehypothesesabove.

RejectthenullhypothesesChapter8Two-TailedTestsinlarge-SamplecaseInlargecase,thegivenconfidencelevelconfidenceintervalofpopulationmeanis:Whenthehypothesestestisgoing,weshouldmakehypothesesforpopulationparameter:Two-tailedtestAstheruleofrejection,wecangetthattheareaofnon-rejectionsamplemeanH0includesallsamplemeanwhichisbetweenstandarddeviationof‘s

and(1)Chapter8Two-TailedTestsinlarge-SamplecaseAsaresult,thenon-rejectionregionofthetwo-tailedhypothesestest’ssamplemeancanbegivenasfollows:Therelationshipbetweenthenon-rejectionregionoftwo-tailedhypothesestestandtheconfidenceregion:IsgivenIsnotgiven(2)Iftheisinthenon-rejectionregiondefinedin(2),supposetheisintheconfidenceregiondefinedin(1).Totheopposite,iftheisintheconfidenceregiondefinedin(1),thesamplemeanwillbeinthehypothesestestnon-rejectionregiondefinedin(2).Chapter8Two-TailedTestsinlarge-SamplecaseSowecangetthestepofhypothesestestcalculatingtheconfidenceregion:Theformofhypotheses:(1)pickoutaconfidenceregionofpopulationmeanwhichisbuiltbysimplerandomsamplefrompopulation:(2)iftheconfidenceregionincludehypothesesvalueof,don’trejectnullhypotheses,otherwise,reject.IsgivenIsnotgivenChapter8Two-TailedTestsinlarge-SamplecaseExample:westillusetheexampleofgolftwo-tailedtest:Fromthesampledatawecanget:TothegivenconfidencelevelWecangettheconfidenceregionwhosepopulationmeanis95%:Thevalueofpopulationhypothesesisintheregion,sowecannotrejectthenullhypotheses.As274.58~282.42Chapter8Two-TailedTestsinlarge-SamplecaseChapter8Testsaboutapopulationmean:small-samplecaseWe’veknownfromtheintervalestimate,whenpopulationobeynormaldistributionandthepopulationvarianceisunknown,thestatisticinthesmall-samplecaseAtthistime,thetestofpopulationmeanshouldbedonebystatistic.Example:iftheaveragescoreofpopulationqualitylevelofairportisequaltoormorethan7,wecanconsidertheservicegivenbytheairportisexcellent.Nowwepickup12passengersrandomlyassample,gettingthequalitylevelscoreofoneairportinLondonasfollows:7、8、10、8、6、9、6、7、7、8、9、8.Assumethelevelofpopulationobeythestandardnormaldistribution,canweconsidertheairportservicequalityexcellent,withthesignificantlevel0.05?1.Establishnullandalternativehypotheses2.Choosestatistict,andcalculateWecanget3.

4.Fromthet-distributiontablewecangetIt’sintherejectionregion,soreject.Andweconsidertheservicesuppliedbytheairportisexcellent.Chapter8Testsaboutapopulationmean:small-samplecaseAttention:insmall-samplecase,thestepsoftestandestimaterulearealmostthesameasinlarge-samplecase,theonlydifferenceissmall-samplecasecorrespondt-distribution,whilelarge-samplecasecorrespondnormaldistribution.Besides,insmall-samplecasewecanestimatebyP-value.Butfort-distributiontableisnotdetailedenough,wecannotgettheexactP-valuefromthetable,buttheprincipleofestimationisthesameasabove.Exercises：P282,T34Ifrejectnullhypotheses.Chapter8Testsaboutapopulationmean:small-samplecaseWesetforpopulationpercentage,referstosomegivenhypothesesvalueofpopulationpercentage,thehypothesestestofpopulationpercentagehasformsasfollows:LefttailedtesttwotailedtestRighttailedtestChapter8Testsaboutapopulationmean:small-samplecaseWeonlyconsidertheconditionof----thehypothesestestwhensamplepercentageobeypopulationpercentageinnormaldistribution.Becausethepercentageisspecialmean,thestepsoftesting