A more modern use of regression analysis is to predict crop yield of grapes (Y) from a cluster count (x). In a grape farm, grape vines start by producing clusters of green berries which eventually mature to be harvested. A count of these clusters can then be used to predict the final crop yield at harvest time. To prove/disprove this, (Y, .x) data was collected at a certain grape farm for 12 years. The data are summarized below. Year 1 2 3 4 5 6 8 9 7 Y (tons/acre) 5.6 3.2 4.5 4.2 5.2 2.7 4.8 4.4 116.37 82.77 110.68 97.50 115.88 80.19 125.24 116.15 117.36 93.31 107.46 122.30 10 11 12 5.4 4.9 4.7 4.1 X The results below are from fitting the above data set to the multiple linear regression model: Y =Bx + Bx +,i=1,2,..., 12; where and are unknown regression coefficients, and 1, 2,...,12 are independent random errors which are assumed to have N (0, 2) distributions. Source of Variation Degrees of Freedom Sum of Squares Mean Sum of Squares F Regression/Model ??? ??? ??? ??? Residual/Error ??? 1.392 12 248.290 Total X x (d) Test the hypotheses: ANOVA Table Variable Estimate of Coefficient 3.263 x 10-2 8.416 x 10-5 Estimates of the Regression Coefficients at the 0.05 level of significance. (a) Complete the above ANOVA table. (b) What proportion of the total variation of the grape yield is explained by the regression model? (c) Test the hypotheses (i) Ho B = 0 versus H (ii) Ho: B=0 versus H: Standard Error 8.697 x 10- 7.799 x 10-5 Ho: B-B=0 versus H: B, 0 for some i = 1,2 ??? B #0 at the 0.05 level of significance; and #0 at the 0.05 level of significance. (5) (3) (5) (10)