Homework #5 2 Linear regression residuals l. (7' points) In lecture, we spent a great deal of time talking about simple linear regression, which you also saw in Data 8. To briey summarize, the simple linear regression model assumes that given a single observation 2:, our predicted response for this observation is 3:; = :90 + 91:33:. (Note: In this problem we write {60, I91} instead of (a, b) to more closely mirror the multiple linear regression model notation.) In Lecture 9 we saw that the 60 = 9;] and 91 2 ll that minimize the average L2 loss for the simple linear regression model are: 90376157 lT 0'3 (a) (2 points) As we saw in lecture, a residual e,- is dened to be the difference between a true response y,- and predicted response y}. Specically, e, = y, 91;. Note that there are a data points, and each data point is denoted by (33, 31,). Show, using the equation for 3;; above, that the average prediction is also the average value of the response, i.e. % 2:21 y,- = $2212] 3J5. (b) (3 points) Show that (1%,?) is on the simple linear regression line. (c) (2 points) Show that the residuals are uncorrelated with the predictor variable, that is where e = l 7.: e- and 0'2 = ._ (e- (3)2. You may assume that at least one 1; 11 I c n. 1 I residual is not exactly zero