Answer the following questions appropriately adhering to all the requirements.

A life assurance company is examining the force of mortality, / . of a particular group of policyholders. It is thought that it is related to the age, x, of the policyholders by the formula: Ux = Bet It is decided to analyse this assumption by using the linear regression model: Y, = a+ B.x; + &; where &; ~ N(0,67) are independently distributed The summary results for eight ages were as follows: Age, X 30 32 34 36 38 40 42 44 Force of mortality, 5.84 4, (x10") 6.10 6.48 7.05 7.87 9.03 10.56 12.66 In ux (3 s.f.) -7.45 -7.40 -7.34 -7.26 -7.15 -7.01 -6.85 - 6.67 [x, = 296 Ex; =11,120 Linux, =-57.129 (linux, )' = 408.50 Ex, In ux =-2,104.5 (i) (a) Apply a transformation to the original formula, /, = Be*, to make it suitable for analysis by linear regression. Hence, write down expressions for Y, a and B in terms of /,, B and c. (b) Plot a graph of In /, against the age of the policyholder, x. Hence comment on the suitability of the regression model and state how this supports your transformation in part (a). [4] (ii) Use the data to obtain least squares estimates of B and c in the original formula. [3] (iii) (a) Calculate the coefficient of determination between In , and x. Hence comment on the fit of the model to the data. (b) Complete the table of residuals and use them to comment on the fit. [5] Age, X 30 32 34 36 38 40 42 44 Residual, e; 0.08 -0.03 -0.06 0.02 0.09 (iv) Calculate a 95% confidence interval for the mean predicted response In //35 and hence obtain a 95% confidence interval for the mean predicted value of //35 . [4] [Total 16]