Please answer the following question:

(a) In your own words, explain how the large-sample properties of maximum-likelihood estimators can help us construct approximate confidence intervals for parameters. [3 marks] (b) A survey of the use of a particular product was conducted in three areas. 100 ran- domly selected potential users were interviewed in each area. Let 91,-, for 2' = 1, 2,3, denote the number of respondents from area 2' who said that they use the product. You want to use a likelihood-ratio test to determine whether the proportion of the population using the product is the same in each of the three areas. i. State the null and alternative hypothesis for this test, and write down the formula for the test statistic. [2 marks] ii. What is the distribution of the test statistic under the null hypothesis? [1 marks] iii. Suppose that the results of the survey were y1 = 38, y2 = 26, and y3 = 29. Compute the value of the test statistic. [5 marks] iv. Would you reject the null hypothesis at the 5% level? What do you conclude? [2 marks] v. Compute a p-value for the observed results. [Hint: remember the link between the chi-squared, Gamma, and exponential distributions.] [3 marks] vi. Draw a simple diagram showing the distribution of the test statistic under Ho, the critical value, the observed value for these results, and the corresponding p-value. [4 marks] You may find the following statistical table helpful: 2 2 2 k Xk0.90 Xk0.95 Xk0.975 Xk099 2.706 3.841 5.024 6.635 4.605 5.991 7.378 9.210 6.251 7.815 9.348 11.345 7.779 9.488 11.143 13.277 hum