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Consider a population in which a proportion U of members have some specified characteristic. Let P denote the corresponding proportion of members in a random sample of size n from the population. (i) Explain clearly why the mean and standard crror of Parc given by F [ P] =0, &.[P] = A (1-A) (3] An insurance company uses a questionnaire to monitor the satisfaction of its customers. In one part customers are asked to answer "yes" or "no" to a particular question. Suppose that a random sample of 200 responses is examined. (ii) Calculate the approximate probability that at least 150 "yes" answers are found in the sample, on the assumption that the true (population) proportion of "yes" answers is 0.7. [4] Suppose the true (population) proportion of "yes" answers (0) is unknown, and for a random sample of 200 responses, the number of "yes" answers is found to be 146. (iii) (a) Calculate an upper (one-sided) 95% confidence interval of the form (D, L) for 0. (b) Calculate a lower (one-sided) 95% confidence interval of the form (L, 1) for 0. (c) A test of the hypotheses: Ho: 0 -0.7 v Hj: D > 0.7 results in a P-value of 0. 198. Comment on how this result relates to the confidence interval in part (iin)(b). 19] ['Total 16]Three insurance company colleagues had just completed an investigation which involved the application of a two-sample f-test to compare two independent samples, each of size 1 1. They were concemed about the validity of the equal variance assumption required for this test. Their data were as follows. A. 21 22 28 27 20 23 26 32 25 21 30 R: 19 18 38 33 24 39 22 20 28 26 30 Ex =275, ExA =7, 033, Exp =297, EIg = 8,559 One of the colleagues suggested a graphical approach for the comparison of the variances. Draw a suitable diagram to represent these data so that the variability of the samples can be compared, and comment briefly on that comparison. [3] (ii) Another colleague suggested using an F-test for the comparison of variances. (a) Perform this F-test at the 5% level to compare the variances and express your conclusion clearly. (b) In addition obtain an approximate value of the P-value for this test hy linearly interpolating between suitable entries in the tables. [6] (iii) The third colleague suggested another procedure using a two-sample f-test in the following way: "For each sample calculate the absolute values of the deviations of the observations from the mean of that sample; then apply a two-sample f-test to the two sets of absolute deviations." (a) Discuss the possible reasoning behind this suggested procedure by considering the potential values of such absolute deviations when the assumption of equal variances is valid and when it is not valid. (b) (1) Calculate the required sets of absolute deviations for the given data. (2) Perform the suggested two-sample f-test at the 5% level stating your conclusion clearly. (3) Obtain, in addition, an approximate P-value for this test by linearly interpolating between suitable entries in the tables. [11] (iv) Comment briefly on the conclusions that may have been reached by the three colleagues. [2] [Total 22]