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7. (a) For any n numbers a1, . .. , an, let a = !) Liz di. Show that for any constant c, it holds that n n [(ai - c)? = >(ai - a)? + n(a- c)? 2=1 i=1 [6 marks] (b) Let {X1, . .., Xn} be a sample from a population with mean / and finite vari- ance 02 > 0. i. Based on the result in (a) above (or otherwise), find the least squares estimator a for u. [3 marks] ii. Is a an unbiased estimator for ? Justify your answer. [3 marks] iii. Compute the mean square error MSE(). Is a a consistent estimator for ? [5 marks) iv. Specify an approximate (1 -a) confidence interval for / when n is large. [3 marks] OLSE 2008/ST102 Page 7 of 11