Answer the questions below with explanations

A general insurance company is debating introducing a new screening programme to reduce the claim amounts that it needs to pay out. The programme consists of a much more detailed application form that takes longer for the new client department to process. The screening is applied to a test group of clients as a trial whilst other clients continue to fill in the old application form. It can be assumed that claim payments follow a normal distribution. The claim payments data for samples of the two groups of clients are (in {100 per year): Without screening 24.5 21.7 35.2 15.9 23.7 34.2 29.3 21.1 23.5 28.3 With screening 22.4 21.2 36.3 15.7 21.5 7.3 12.8 21.2 23.9 18.4 (i) (a) Find a 95% confidence interval for the difference between the mean claim amounts. (b) Comment on your answer. [6] (ii) (a) Find a 95% confidence interval for the ratio of the population variances. (b ) Hence, comment on the assumption of equal variances required in part (i). [4] (iii) Assume that the sample sizes taken from the clients with and without screening are always equal to keep processing easy. Calculate the minimum sample size so that the width of a 95% confidence interval for the difference between mean claim amounts is less than 10, assuming that the samples have the same variances as in part (i). [3]2 You are given the following data in respect of a certain group of lives: ET 60 6 387 61 24 766 62 34 1.093 63 20 699 64 9 339 Test the hypothesis that the mortality rates of the above lives are equal to the following se which are taken from a certain standard table: Age r 60 0.02287 61 0.02525 62 0.02778 63 0.03049 64 0.033393 You are comparing the mortality of a certain group of young policyholders with that of a standard table, the central death rates of which are denoted by {m, }. You have compiled the following table: r E-mr VEams 9 11.7 -0.79 6 6.8 -0.31 9 10.2 -0.76 09 10 9 11.6 -0.76 12 13.9 -0.51 19 15.8 +0.81 22 17.1 +1.18 30 17.4 +3.02 8 20 19.3 +0.16 18 17.8 +0.05 10 23 16.6 +1.57 11 21 14.1 +1.84 12 21 13.3 +2.11 13 15 12.9 +0.58 14 11 12.2 -0.34 15 9 11.2 -0.66 16 8 9.9 -0.60 17 8 8.7 -0.24 18 -J 7.2 -0.07 19 6 7.4 -0.51 20 5 6.0 -0.41 21 4 5.4 -0.60 22 3 3.3 -0.17 23 3.7 -1.40 - N 24 3.1 -1.19 Test the goodness of fit of the standard table