(2) Let X1,..., X, denote independent and identically distributed normal random variables with expectation / and variance o'. (a) Assume n is even and consider the estimator X defined below where the even indexed samples are given twice the weight of the odd indexed samples: n/2 = 3n (X2i-1 + 2X2i). i=1 (i) Show that X is unbiased in estimating /. [4 marks] (ii) Prove that Var(X) = o' and that X is a consistent estimator for (. [4 marks] (iii) Derive a 100(1 - a)% confidence interval for / based on the estimator X, assuming that o' is known. Clearly state any distributional results that you are using and carefully define any notation used. [4 marks]