55 The following variables were recorded for several counties in Minnesota in 1977: y = average rent paid per acre of land with alfalfa x1 = average rent paid per acre for all land x2 = average number of dairy cows per square mile X3 = proportion of farmland in pasture The data for 34 counties are given in Table 7.5 (Weisberg 1985, p. 162). Can rent for alfalfa land be predicted from the other three variables? (a) Find and s?. (b) Find , and Bo using Sxx and Syx as in (7.46) and (7.47). (c) Find R2 and R. SC X'X , n-1 (7.44) X'y Syx = (7.45) n-1 Note that X'y in (7.45) contains terms of the form {i=1 (Xij X;)y; rather than = (Xij X;)(y; 5) in (7.43). It can readily be shown that (Xij X;)ly; - y) = 2; (Xij X;)y: (see Problem 6.2). From (7.37), (7.44), and (7.45), we have as B = (n 1)(XX.)-1 Xy n-1 (2) XY Silber (7.46 and from (7.38) and (7.46), we obtain Bo = 4 8x =y- ss. (7.47) TABLE 7.5 Land Rent Data X1 X2 X3 X1 X3 18.38 20.00 11.50 25.00 52.50 82.50 25.00 30.67 12.00 61.25 60.00 57.50 31.00 60.00 72.50 60.33 49.75 15.50 22.29 12.36 31.84 83.90 72.25 27.14 40.41 12.42 69.42 48.46 69.00 26.09 62.83 77.06 58.83 59.48 17.25 18.51 11.13 5.54 5.44 20.37 31.20 4.29 8.69 6.63 27.40 31.23 28.50 29.98 13.59 45.46 35.90 .24 .20 .12 .12 .04 .05 .27 .10 .41 .04 .12 .08 .21 .17 .05 .16 .32 8.50 36.50 60.00 16.25 50.00 11.50 35.00 75.00 31.56 48.50 77.50 21.67 19.75 56.00 25.00 40.00 56.67 9.00 20.64 81.40 18.92 50.32 21.33 46.85 65.94 38.68 51.19 59.42 24.64 26.94 46.20 26.86 20.00 62.52 8.89 .08 23.81 .24 4.54 .05 29.62 .72 21.36 .19 1.53 .10 5.42 .08 22.10 .09 14.55 .17 7.59 .13 49.86 .13 11.46 .21 2.48 31.62 .26 53.73 .43 40.18 15.89 .05 .10 .56 55 The following variables were recorded for several counties in Minnesota in 1977: y = average rent paid per acre of land with alfalfa x1 = average rent paid per acre for all land x2 = average number of dairy cows per square mile X3 = proportion of farmland in pasture The data for 34 counties are given in Table 7.5 (Weisberg 1985, p. 162). Can rent for alfalfa land be predicted from the other three variables? (a) Find and s?. (b) Find , and Bo using Sxx and Syx as in (7.46) and (7.47). (c) Find R2 and R. SC X'X , n-1 (7.44) X'y Syx = (7.45) n-1 Note that X'y in (7.45) contains terms of the form {i=1 (Xij X;)y; rather than = (Xij X;)(y; 5) in (7.43). It can readily be shown that (Xij X;)ly; - y) = 2; (Xij X;)y: (see Problem 6.2). From (7.37), (7.44), and (7.45), we have as B = (n 1)(XX.)-1 Xy n-1 (2) XY Silber (7.46 and from (7.38) and (7.46), we obtain Bo = 4 8x =y- ss. (7.47) TABLE 7.5 Land Rent Data X1 X2 X3 X1 X3 18.38 20.00 11.50 25.00 52.50 82.50 25.00 30.67 12.00 61.25 60.00 57.50 31.00 60.00 72.50 60.33 49.75 15.50 22.29 12.36 31.84 83.90 72.25 27.14 40.41 12.42 69.42 48.46 69.00 26.09 62.83 77.06 58.83 59.48 17.25 18.51 11.13 5.54 5.44 20.37 31.20 4.29 8.69 6.63 27.40 31.23 28.50 29.98 13.59 45.46 35.90 .24 .20 .12 .12 .04 .05 .27 .10 .41 .04 .12 .08 .21 .17 .05 .16 .32 8.50 36.50 60.00 16.25 50.00 11.50 35.00 75.00 31.56 48.50 77.50 21.67 19.75 56.00 25.00 40.00 56.67 9.00 20.64 81.40 18.92 50.32 21.33 46.85 65.94 38.68 51.19 59.42 24.64 26.94 46.20 26.86 20.00 62.52 8.89 .08 23.81 .24 4.54 .05 29.62 .72 21.36 .19 1.53 .10 5.42 .08 22.10 .09 14.55 .17 7.59 .13 49.86 .13 11.46 .21 2.48 31.62 .26 53.73 .43 40.18 15.89 .05 .10 .56