(a A random sample of 30 coffee drinkers were each asked to taste-test a new brand of coffee. The responses are listed below with L representing 'like' and D representing 'dislike'. D D D L D L A cafe manager claims that more than half of all the coffee drinkers like this new brand of coffee. (i) Test at 1% significance level for the manager's claim. (7 marks) (ii) Estimate the p-value. (2 marks) Let X1, X2,..., X, are independent random samples from a population with the following probability density function: f(x; 0) = 4xe , for x > 0 0, elsewhere Given E(X) = 40 and Var(X) = 402 . If 0 = X is an estimator of 0, 4 (i) show that the estimator 0 is unbiased. (3 marks) (ii) find Var(@). (4 marks) (iii) show that the estimator O is an efficient estimator (MVUE) of 0. (7 marks) (iv) determine whether the estimator O is consistent for 0. (2 marks)