Question

4. (a) Let Xi,X ,x, be n observations from an N(u2) distribution, and define the estimators (i) Determine whether T and T2 are unbiased estimators of u. 4 points (ii) Compute the variances Var(Ti), and Var(T2). Which is the better estimator T or T2 -and why? [2 points] Determine the maximum likelihood estimator of μ. (iii) [5 points) (b) A manufacturer is testing the performance of two products, A and B. At each of 20 field sites, product A and product B are tested together under identical conditions. At each site it is recorded which product, A or B, performed better. Of the 20 sites, 12 judged product B to have the better performance. Perform a statistical hypothesis test to determine whether product B performs better than product A.

Answer 1

a) Let X1, X2, , xn be n observations from an N(r, 2) distribution, and define the estimator X1 + X2++Xn-1- 2X, T2(x) = ぬ To check whether it is an unbiased estimator To check whether it is an unbiased estimator Note that μ--(X1 + X2 + + Xn) Note that μ--(X1 + X2 +--+ X.) xi + + +ぬ-,-28, Using linearity property of expectation Using linearity property of expectation Thus X1 + X2 ++X-1 - 2X, Thus T2(x) = X. is an unbiased estimator is not an unbiased estimator

1st,jsn 1si,jsn 1si.jsn 1si.jsn Now for any fixed j, j, since X,X, are independent we find Thus Var[T1(X)]- 1st.jsn 1si.jSn 1si.jSn 1si.jsn 1si.jSn 1SİJSn

+4.1-24)21-u) EI(x, + x, + W. 1sisn-1 1sisn-1 15i.jSn-1 [(n-1)2(u*) + 4μ 2-4(1-1):12)]-μ2 (n - 3)2 (n - 3)2 (n - 3)2 (n -3)2 Thus T2 (X) is a better estimator for the mean