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3. (a) A random sample {X1, X2, , Xn} is drawn from the following probability distribution: f(x;0) = 0 forAi. Derive the maximum likelihood estimator of 0 and explain why this is a maximum. (6 marks) ii. If n = 4, such that T1 = 0.70, x2 = 0.92, x3 = 0.21 and x4 = 0.34, determine the maximum likelihood estimate of 0. (2 marks) iii. As n 00, would you expect the maximum likelihood estimate of 0 to get closer to 0? Briefly explain your answer. (2 marks) (b) A random sample of size n = 9 drawn from a normal distribution had a sample variance of s2 = 72.76. Construct a 90% confidence interval for 02. (5 marks) (c) Let X be a random variable such that X ~ Exp() Suppose we wish to test the null hypothesis H0 : l = 2 VS. H1 : l > 2. Consider the test procedure which rejects H0 if X < 0.75. i. Compute the probability of a Type I error, expressing your answer in terms of the exponential constant e. (2 marks) ii. Determine the power function of the test, expressing your answer in terms of the exponential constant e. (3 marks) 4. (a) Suppose that you are given independent observations y1, y and Y such that: y1=3a+B+E1 = y2=a-28+82 - y = =-a+2B+E3. The random variables Ei, for i = 1,2,3, are normally distributed with a mean of 0 and a variance of 4. i. Find the least squares estimators of the parameters a and B, and verify that they are unbiased estimators. (7 marks) ii. Calculate the variance of the estimator of a. (3 marks) (b) A supermarket chain investigated its profit margins over a period of 4 years. It splits its merchandise into 6 categories and recorded the profit margin for each category for each year. The investigators constructed an ANOVA table, which is given below (some entries are missing). Source DF SS MS F Category 62.10 Year 3.31 Error 3.14 Total i. Complete the table in your answer booklet. (6 marks) ii. Is there a significant difference between the profit margins of different categories? Use a 5% significance level. (4 marks)

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