Lecture16

# Lecture16 - Lecture15 Stat3507.5 TwoPopula>onMeans...

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Unformatted text preview: Lecture15 Stat3507.5 ConfidenceIntervalsfor TwoPopula>onMeans LookingAhead... Today: ComparingTwoPopula>onMeans Topic1:LargeSamples Topic2:SmallSamples Topic3:PairedData WednesdayMarch2,2011 Lab3onConfidenceIntervals Duedate:FridayMarch11,2011 FridayMarch4,2011 Quiz[bringyourcalculators] 7.3:Comparingtwopopula>ons(largesamples) Examples: Domenhaveahighermeanmetabolismthan women? IsthemeanIQofPurduesimilartothemeanIQ ofIndianaU.? DoDuracellAAbaZerieslastlongerthan EnergizerAAbaZersonaverage? Nota>onfortwopopula>ons Population Population mean Population standard deviation 1 2 1 2 1 2 TwoSampleSe]ngs TwoSRSfromtwopopula>ons: SRSofsizen1isselectedfromthe1stpopula>on SRSofsizen2isselectedfromthe2ndpopula>on n1 andn2arebothlarge(30+),andcanbeofdifferentvalues Population Sample size Sample mean Sample standard deviation 1 2 n1 n2 s1 s2 Example4 Frompage307,wewanttocomparethelife>me (inhours)oftwobrandsofbaZeries: Whichbrandhasalongerlife>me?Byhow much? Whatsta>s>careyouusing? Whatistherelevantparameter? SAMPLINGDISTRIBUTION FORTHEDIFFERENCEOFMEANS Themean: Thestandarddevia>on: C.L.T.atworkagain Thesamplingdistribu>onofthedifferenceof meansisnormalifbothsamplesizesarelarge So, Howtostandardize? Recall, Similarly, Whenn1andn2arelarge,wecanreplacethe's withs's. TWOSAMPLECONFIDENCEINTERVAL FORTHEDIFFERENCEOFMEANS ThisCIisvalidforalldistribu>onsforthetwo popula>onswhenn1andn2arebothlarge Example4(cont'd) Frompage307,wewanttocomparethe life>me(inhours)oftwobrandsofbaZeries: Constructa95%confidenceintervalforthe differencebetweenthetwobrandsof baZeries. Again,makesureyoucaninterprettheinterval! Twosampleconfidenceintervalforthe differenceofmeans Zcrit(95%,doublesided)=? Twosampleconfidenceintervalforthe differenceofmeans Zcrit(95%,doublesided)=? 0.380.69 Ex32onpage312 !me beginning end Sample size 40 40 Sample mean 143.7 131.7 s 21.2 20.9 Find95%confidenceintervalforthetruemeandifference Ex32onpage312 !me beginning end Sample size 40 40 Sample mean 143.7 131.7 s 21.2 20.9 Find95%confidenceintervalforthetruemeandifference Interpretthe95%CI Yes,thereisevidencesugges>ngthatwhen thequizzesaretakenaffectsthetruemean scores.Infact,wewouldes>matethat studentstakingquizzesatthebeginningof classwouldscore,onaverage,between3and 21pointshigheronthefinalexam. Weare95%confidentinthisconclusion. 7.5Twosampletconfidenceintervals Atleastonesamplebeingsmall Twoindependentsamples FormulaforCI: df n 1 What is the degree of freedom? DegreeofFreedom df = (s (s 2 1 2 1 n1 + s n 2 ) 2 2 2 2 2 n1 ) n1 -1 (s + 2 2 n2 ) n 2 -1 Ifthedfisnotaninteger,rounddowntobe conserva>ve(I.E.df=9.86,use9) Example6 Metabolismratesof12randomwomenand 7randommenweremeasured. Finda95%CIforthedifferenceinmean metabolismbetweenmenandwomen Solu>on 364.0(tcrit)*89.77 Needtocalculatedegreeoffreedomtofindtcritval DegreeofFreedom df = (s (s 2 1 2 1 n1 + s n 2 ) 2 2 2 2 2 n1 ) n1 -1 (s + 2 2 n2 ) n 2 -1 Df=12.6rounddown,usedf=12 So, t cri!cal value (two sided 95% CI, df=12) = 2.179 Solu>on 364.02.179*89.77 =364.0195.6=(559.6,168.4) ACIfromPairedData Osen>mes,dataiscollectedinpairswhich createstheillusionoftwosamples,althoughin realitythereisreallyonlyonesample. Example--examsscoresforSTAT350 Obs Exam1 Exam2 1 74 87 2 89 86 3 83 79 ... Whyisthisconsideredonlyonesample? PairedData--treatlikeonesample Takethedifferenceofthescores,thenstudythe "differences"asasinglesample Obs Exam1 Exam2 Difference 1 74 82 8 2 89 86 3 3 83 79 4 ... Findthemeanofthedifferences. Findthestandarddevia>onofthedifferences. Example7 Supposeasampleofn studentsweregivenadiagnos>c testbeforeandagaincomple>ngamodule.Herewehave n1=n2=10students,thedataisasfollows: Finda95%confidenceintervalforthedifferences. Solu>on: Paired Data: n1=n2=10 Finda95%CIforthedifferences. Degreeoffreedom=9 Tcri>calvalue=2.262 Solu>on: Paired Data: n1=n2=10 Finda95%confidenceintervalforthedifferences. Degreeoffreedom=9Tcri>calvalue=2.262 Mean =1.600000 Std Devia!on = 2.95146 95%CI:1.6+/2.262*2.95/sqrt(10) =(1.62.1,1.6+2.1)=(0.5,3.7) ...
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