Let (X1, Yi), (X2, Y2), .... (Xn, Ym) ~ Na(u, E) be bivariate normal observations with covariance matrix E = [01 0102P] 0102p This question is concerned with inference for the unknown correlation coefficient p. Let s* and sy denote The sample variances of the X, and Y, respectively, and let sxy denote the sample covariance between the X and Yr. An obvious estimator for p is the Pearson sample correlation coefficient SXY SXBy We consider the Fisher-Z transformation given by z = arctanh(r) = log and you can assume that, asymptotically, z ~ N arctanh(p), _ (a) Show that T = n -3(z - arctanh(p)) is a pivotal quantity (asymptotically) for p. [2 marks] (b) Using the pivotal quantity from Part (a), construct a symmetric (1 - o)100% confidence interval for p. [4 marks] (c) On MyUni, you will find a . cav file called bivariate_normal_data. cav. This data contains 100 observations from a bivariate normal distribution. Using this data, construct a 95% confidence interval for the true correlation p between X and Y. Round your confidence interval to 3 decimal places where appropriate, and provide any R code used in your calculations. [6 marks] (d) Using your confidence interval from Part (c), test the hypothesis Ho : P = at the o = 0.05 level of significance. Be sure to give a precise conclusion of your findings