Question 4 (20 marks) Consider a full rank linear model y = X8+ with the usual assumptions required for inference. The variance of e is denoted as o2 and the dimensionality of is denoted as p. This question derives the prediction interval for the difference of the responses of two independent future observations with predictors X1 and x2 respectively. If b is the least squares estimator of B, the prediction interval uses the unbiased point estimator, (x1 - x2)+b, of the mean difference of the two independent future responses. The prediction error is the difference of the two future observations minus the unbiased point estimator, i.e. y1 - y2 (x1 - x2)+b. (a) Calculate the variance of the prediction error. (5 marks] (b) Show that the prediction error is independent of SS Res. [5 marks] (c) Derive a t-distributed quantity based on the prediction error and state its degrees of freedom. (5 marks] (d) Thus write down a formula for a 100(1 a)% prediction interval for the difference of future responses at xi and X2 [5 marks]