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Question 3 A company operates a drinking water purifying process at two different sites and there is interest in examining the purity of the water produced. Data have been collected from each site over n weeks. The data are denoted by {(x1, 71), ..., (Xn, ))), where x; and y; denote a measurement of purity, on a scale between 0 and 1, at sites 1 and 2 respectively, in week i. An appropriate probability model is given by the density function f(x, y) = BoxB-16-1 where 0 1. The two parameters f and 6 determine the marginal distributions of purity at sites 1 and 2 respectively. i) Write down the joint likelihood function for B and 6. Use this to construct the log-likelihood function. Hence find the maximum likelihood estimators of B and 5 in terms of the observed values X1, ...,*n and y1, ...,In and construct the Hessian matrix. ii] Given that n = 30, Et=, log. (x;) = -1.79 and )=1 log. ()i) = -1.63 evaluate the maximum likelihood estimates of B and 6. Use these estimates along with the inverse of the sample information matrix In (8, 6) to construct an approximate 95% confidence interval using the Wald method. iii) Comment on what this confidence interval tells you about the evidence for different purity distributions at sites 1 and 2