3. (i) You are asked to verify the adequacy of the following graduation of mortality experience obtained for a particular type of policy. Age Exposed to risk Actual deaths. d,c Graduated mortality,q,.c Expected deaths 65 60,000 1 .370 0.0225 1350 66 50,000 1 .200 0.0250 1250 67 43,000 1 .200 0.0276 1 187 68 37,000 1.090 0.0304 1125 69 30,000 1.010 0.0334 1002 70 26,000 950 0.0368 957 71 23,000 980 0.0406 934 72 21 ,000 950 0.0449 943 73 20,000 1 .000 0.0497 994 74 1 8,000 960 0.0551 992 (a) (5 marks) Comment on the graduation smoothness. (b) (11 marks) Check the goodness of fit using the standardised deviation test. (ii) (3 marks) A graduation covers 15 age groups reveals 5 positive and 10 negative devi- ations. Using the sign test, comment on the goodness of fit for the considered graduation. (iii) [2 marks] You perform graduation and fit a function to mortality data. You observe that for a function using 4 parameters your residual deviance is 23.53. whereas considering a more complicated function containing 6 parameters reduces the residual deviance up to 15.51. ls considering the function with 6 parameters justified? (iv) [4 marks] A 10,000 population is observed over a three year period. During this period 1,000 deaths are observed, however the population size remains fairly constant. Assuming the deaths in this population can be modelled using a Poisson distribution. find the maximum likelihood estimate for this population. Using this, estimate the 95% confi- dence interval for the force of mortality over this period. Total: 25 marks