B. In addition, for Problem 6 in Part A (ii) above, do the following exercises: (i) Fit a least squares regression line to the data with y = output and x = 1/(wind speed). (ii) Produce all relevant plots of the residuals (as discussed in slides 19-20 of Chapter 1A lecture notes) from your fitted linear model in part (i) above and use them to comment on whether the model assumptions (i.e. linearity of the regression curve E(Y X ), equal error variances and normality of the errors c's) seem reasonable. (Note: you must present all five plots discussed in the notes.) (iii) Find the coefficient of determination R" and interpret it. (iv) Find a 99% confidence interval (CI) for the slope (B1) of your linear model fitted in part (i) above. (v) Find a 95% confidence interval for the average output when the wind speed is 3.2. vi) Suppose the wind speed at a particular windmill is 9.05. Find a 95% prediction interval for the output of this mill, i.e. an interval in which you are 95% sure the output of this windmill will be. C. Recall the derivation of the least squares regression line shown in slides 12-14 of Chapter 1A notes. (i) Complete the steps outlined in those slides and show that the slope (B1) and intercept (Bo) estimates for the least squares regression line are given by: B1 = Li=(Ti - x) (yi - y) Eil(xi -x)2 and Bo = y - Bix, where all notations have their standard meanings as used in the notes. (Hint: in the final step, to arrive at the desired expression of 1 as above, you may need to use the fact that Li (i -x) = 0 and _1 1(yi - y) = 0, both of which follow quite easily from the definitions of a and y.) (ii) Further, let p denote the sample correlation coefficient between Y and X (defined formally below), and let of = m Zi=(yi -y) and ox = " Et(xi-x)2. (These notations are also defined in slides 41-43 of Chapter 1A notes.) Then, prove that 1 from above also satisfies the following form: B1 = Pox' where P = EN(xi - x) (yi - y) VEL(xi -x) VEiz(yi - y)2' by = Vor and ox = Vox, with of and ox as above. What does this result tell us about the nature of the relationship between B1 and p? And also between ? and p2 (which is the same as R2, the coefficient of determination)? (Note: For all proofs above, you must show in detail all your steps in arriving at the final conclusions.)