3 . Show that the hat matrix is symmetric and idempotent. (That is, show that H' = H and that H H = H.) . Show that the vector of observed residuals can be expressed as e = (I H )6. . Prove or disprove: y'g) = y' y. (a) Show that in the simple linear regression setting, the usual F test statistic of H0: ,81 = 0 can be written as R2 2 F 2 L 1 R2 (b) In a simple linear regression with n = 300, how large must R2 be in order for there to be signicant evidence against the null hypothesis from a), at the 5% signicance level. . Suppose we run a simple linear regression and nd the estimated covariance matrix of the parameter estimators is: 13.4 -62.0 -62.0 308.5 Use this covariance matrix to nd the estimated variance of the estimated mean of y when a: = 8. What is the estimated variance? (Note that the estimated mean of y when x = 8 is ,80 + ,31 - 8.)