4. Recall the derivation of the least squares regression line shown in slides 1214 of Chapter 1A notes. (a) Complete the steps outlined in those slides and show that the slope (31) and intercept (BO) estimates for the least squares regression line are given by: 51 _ Lina-(mE) _ 1y=1(a:,-:)2 and 50 y'lx, where all notations have their standard meanings as used in the notes. (Hint: in the nal step, to arrive at the desired expression of [3'1 as above, yOu may need to use the fact that 23:1(m, 5T:) = 0 and Z:=1(y,- 37) = 0, both of which follow quite easily from the denitions of :7: and 37.) (b) Further, let ,6 denote the sample correlation coeicient between Y and X (dened formally below), and let (3?; = ZLJy, m2 and (3% = zzlci :3)? (These notations are also dened in slides 4143 of Chapter 1A notes.) Then, show that 61 from above also has the following form: 6V A 7-b 332' 53 'i _ A A A A 51 = A , where [5 = W, 01/ = 0%; and 0X = 0'?\" 0X V i=1(37i 3?) i=1(yi y) with (\"TY and 6% as above. What does the result say about the nature of the relationship between 61 and ,6? And also between ,8? and ,62 (which is the same as R2, the coefcient of determination)? (Note: For all proofs above, you must show in detail all your steps in arriving at the nal conclusions.)