Lecture18

Lecture18 - STAT350Lecture18 8.1HypothesesandTestProcedures...

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Unformatted text preview: STAT350Lecture18 8.1HypothesesandTestProcedures 1 HypothesesTes;ng ConfidenceIntervals Es#mateanunknownpopula;onparameter. givearangeofplausiblevaluesofthetrueparameter HypothesisTests Decisionproblem Assesstheevidenceprovidedbythesampledatainfavorof oragainstthenullhypothesis Sta;s;calhypothesesarealwaysaboutthepopula;on parameters,notsamplesta;s;cs. 2 NullandAlterna;veHypotheses H0 Thenull hypothesisistheasser;onthatis ini;allyassumedtobetrue. Ha Thealterna#ve hypothesisistheclaim thatcontradictorytoH0 Next: examples on se1ng up the hypotheses Later: how to test the hypotheses 3 Example(ex3of8.1) Manyolderhomeshaveelectricalsystemthat usefusesratherthancircuitbreakers.A manufacturerof40ampsfuseswantstomake surethatthetrueaverageamperageatwhich itsfusesburnsoutisindeed40.Whatnull andalterna;vehypotheseswouldbeof interesttothemanufacture? 4 Example(ex3fromch8.1) Manyolderhomeshaveelectricalsystem thatusefusesratherthancircuitbreakers. Amanufacturerof40ampsfuseswantsto makesurethatthetrueaverageamperage atwhichitsfusesburnsoutisindeed40. H0:=40vs.Ha:40 5 MoreExamples Translatethefollowingresearchques;onsinto appropriatehypotheses. CensusBureaudatashowthatthemean householdincomeintheareaservedbya shoppingmallis$62,500peryear.Amarket researchfirmques;onsshoppersatthemall tofindoutwhetherthemeanhousehold incomeofmallshoppersishigherthanthat ofthegeneralpopula;on. 6 MoreExamples Translateeachofthefollowingresearchques;onsinto appropriatehypotheses. Lastyear,yourcompany'sservicetechnicianstook anaverageof2.6hourstorespondtotroublecalls frombusinesscustomerswhohadpurchased servicecontracts. A)Dothisyear'sdatashowadifferentaverage response;me? B)Dothisyear'sdatashowahigheraverage response;me? C)Lowerresponse;me? 7 TestofSignificance Thetestofsignificanceisdesignedtoassessthe strengthoftheevidence from data againstthenull hypothesis. TheNullHypothesis: Rejectthenullhypothesisonlyifsampleevidence suggeststhatH0isfalse WhatifwearewrongTypeIError Donotrejectthenullhypothesisifthesampledoes notstronglycontradictH0 Wewillcon;nuetobelieveinthetruthofH0 WhatifwearewrongTypeIIError 8 Test of Hypotheses Amethodforusingsample data todecide betweenthetwocompe;nghypothesesunder considera;on Consider: Whatdoesthehypothesissayabouttheparameter? Whatdoesthesamplesuggestabouttheparameter? Dotheyagreeordisagree? Theresultsofatestmeasureshowwellthedata andthehypothesisagree. 9 example Howmanycreditcardsdopeoplehave? Supposewehaveasampleofsize100.The samplemeanis4andthesamplestandard devia;onis2.Constructa95%C.I.for. 41.96(2/sqrt(100)) [3.608,4.392] Ifsomeoneclaimsthat=2. Whatdoyouthink? 10 example(cont.) SetupourHypotheses: Let'sassumeH0istrue Recall,samplemean=4 If=2,isasamplemeanof4evenpossible? Whatistheprobabilityofsamplemean=4 assumingthenullistrue? 11 example(cont.) AssumeH0:=2istrue: Findtheprobabilityofgekngasampleof100 withameanof4ormoreandSD=2. 12 Calcula;ngthePvalue AssumingH0istrue(=2) Usingthesamplingdistribu;onof P(X 4) = P(Z 10) = 0 13 Pvalue Alsocalledtheobserved significance level PValue: istheprobability,computedassumingthatH0istrue,that thetest sta#s#cswouldtakeas extreme or more extreme valuesastheoneactuallyobserved. ThesmallerthePvalue,themorecontradictoryisthedata toH0. H0shouldberejectedifthePvalueissufficientlysmall Credit Card Example:Pvalue=? 14 MoreaboutPvalue... WhenthePValueissmall,thereare2 possibili;es: 1--Thenullhypothesisistrueandourobserved effectisextremelyrare! Type I Error ORmorelikely... 2--Thenullhypothesisisfalseandourdatais tellingusthisbythesmallPvalue! 15 Significancelevel: Acutoff point(decisivevalue)thatweusetocompare ourPvalueto,andmakeadecision howmuchevidencedoweneedtorejectH0 ? Thiscutoffpointisthesignificancelevel. denoted . Itisannouncedinadvance servesasastandardonhowmuchevidenceagainst H0weneedtorejectH0. Typicalvaluesof:0.05, 0.01 (some#mes 0.10). Ifnotstatedotherwise,assume=0.05. 16 Example 1. Inarandomsampleof100lightbulbs,7are founddefec;ve.Isthiscompa;blewiththe manufacturer'sclaimofonly5%ofthelight bulbsproducedaredefec;ve?(Use=0.01) H0:=5%vs.Ha:>5% IfH0istrue,whatisthesamplingdistribu;onofthesample propor;onp? 17 Solu;on H0:=5%vs.Ha:>5% =0.01 IfH0istrue,thenpisapproximatelynormalwith mean0.05andstandarderror0.02170.02(CLT) Standardizep: Z=(p0.05)/0.02z=1whenp=7% Pvalue=P(p>0.07)=P(z>1)=10.8413=0.1587> DonotrejectH0 18 Sta;s;calSignificance WhenPvalue ,wesaythatthedataare sta;s;callysignificantatlevel DatawithaPvalueof0.02are sta;s;callysignificantatlevel0.05, butNOTsta;s;callysignificantatlevel0.01 muststatethesignificancelevelbefore tes;ng. 19 Theconclusion/decision: Guidelines: IFpvalue<RejectH0 IFpvalue>FailtoRejectH0 AlwaysreportaPvaluewithyourconclusionand writethe conclusioninlayman'sterms. 20 ProceduresforHypothesisTes;ng 1. StateHypotheses 2. Determineanappropriatelevel. 3. Calculatetheappropriateteststa;s;c creditcardexample,itwastheZ=20. UseZforlargesamplemeanorpropor;on 4. FindthePvalue 5. RejectH0orFailtoRejectH0 6. Stateaconclusioninlayman'sterms 21 Example8.4 Therecommendeddailydietaryallowance (RDA)forzincamongmalesolderthan50years oldis15mg/day.Anar;clereportedthe followingdataonzincintakeforasampleof malesage6574years: n = 115, X = 11.3,s = 6.43 Doesthisdatasuggestthat,theaveragedaily zincintakefortheen;repopula;onofmaleage 6574,islessthantheRDA? 22 ...
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