Nordheim Statistics 411 Spring 2015 (Apr 14 or 16) Ratio Estimation with Strata We wish to estimate the mean number of worker-hours lost for a company in the current year. We have complete data from the previous year. The company has two plants: A and B. Plant A has 1000 workers and the total number of hours lost in the previous year was 16300. Plant B has 1500 workers and the total number of hours lost in the previous year was 12800. The two plants are viewed as strata. An SRS with n=10 was obtained for each plant. xa=hours lost previous year – plant A ya=hours lost current year – plant A xb=hours lost previous year – plant B yb=hours lost current year – plant B xa ya xb yb 12 13 10 8 24 25 8 0 15 15 0 4 30 32 14 6 32 36 12 10 26 24 6 0 10 12 4 2 15 16 0 4 0 2 8 4 14 12 16 8 > tauxa=16300 > tauxb=12800 > Na=1000 > Nb=1500 > na=10 > nb=10 > muxa=tauxa/Na > muxb=tauxb/Nb > mxa=mean(xa) > mxb=mean(xb) > mya=mean(ya) > myb=mean(yb) > N=Na+Nb 0 5 10 15 20 25 300 5 10 15 20 25 30 35hours_prevhours_currplantAplantBLet us first consider using separate ratio estimation: > Bhata=mya/mxa > Bhatb=myb/mxb > Bhata > Bhatb [1] 1.050562 [1] 0.5897436 > ybarhatrs=(Na/(N)*Bhata*muxa+(Nb/(Na)*Bhatb*muxb > ybarhatrs [1] 9.86915 > resars=ya-Bhata*xa > resbrs=yb-Bhatb*xb > srsqars=(sum(resars**2))/(na-1) > srsqbrs=(sum(resbrs**2))/(nb-1) > srsqars > srsqbrs [1] 3.477065 [1] 9.726934 > vhatrs=(1-na/Na)*((Na/N)**2)*srsqars/na+(1-nb/Nb)*((Nb/N)**2)*srsqbrs/nb > vhatrs [1] 0.4029119 Now consider combined ratio estimation: > xbarst=(Na/N)*mxa+(Nb/N)*mxb > ybarst=(Na/N)*mya+(Nb/N)*myb > Bhat=ybarst/xbarst > Bhat [1] 0.8677966 > mux=(tauxa+tauxb)/N > ybarhatrc=Bhat*mux > ybarhatrc [1] 10.10115 > resarc=ya-Bhat*xa > resbrc=yb-Bhat*xb > vresarc=var(resarc) > vresbrc=var(resbrc) > vhatrc=(1-na/Na)*((Na/(N))**2)*vresarc/na+(1-nb/Nb)*((Nb/(N))**2)*vresbrc/nb > vhatrc [1]
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