> vy=var (y) > vz=var (z) > c(vx, vy, vz) [1] 144418.27 225582.61 14015.52 > > reg=1m (y^x) > summary(reg) Call: 1m (formula = y x) Residuals: Min -147.179 1Q Median 3Q Max -77.542 -8.385 53.611 239.163 Coefficients: Estimate Std. Error t value Pr(>|t|) 102.16744 -0.70 0.09589 (Intercept) -71.51136 1.21246 0.5 12.64 1.78e-07 *** Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 1 Residual standard error: 120.9 on 10 degrees of freedom Adjusted R-squared: 0.9352 Multiple R-squared: 0.9411, F-statistic: 159.9 on 1 and 10 DF, p-value: 1.784e-07 > mse=sum(reg $residuals^2)/10 > mse [1] 14606.97 (a) Estimate using the sample mean . (b) Estimate using the ratio estimator. (c) Estimate using the regression estimator. (d) Compute the (approximate) relative efficiency of the ratio estimator with respect to the regression estimator. 3. A corporation is interested in estimating the mean earnings from sales of color television sets at the end of a three-month period. The total earnings figures are available for all districts within the corporation for the corresponding three-month period of the previous year. A simple random sample of 12 district offices is selected from 123 offices within the corporation. Use the data in the accompanying table to estimate the mean earnings for offices within the corporation and place a bound on the error of estimation. Take Tx = 128,200. Office 1 Previous year xi Current year yi 550 610 2 720 780 3 1500 1600 4 1020 1030 5 620 600 6 980 1050 7 928 977 8 1200 1440 9 1350 1570 10 1750 2210 11 670 980 12 729 865 You may find the following R codes and outputs helpful. > x=c (550,720, 1500, 1020,620,980,928,1200,1350,1750,670,729) > y=c(610,780, 1600, 1030, 600, 1050,977, 1440, 1570, 2210,980,865) > > f=1-12/123 > mux=128200/123 > c(f,mux) [1] 0.902439 1042.276423 > mx=mean(x) > my=mean(y) > r=my/mx > z=y-r*x > mz=mean (z) > c(mx, my, mz,r) [1] 1.001417e+03 > vx=var(x) 1.142667e+03 -8.526513e-14 1.141050e+00