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(i) Stating clearly any assumptions you make, carry out statistical tests to determine whether, for the population being studied: (a) individuals under 40 watch more television on average than those over 40 (b) the average number of hours of viewing amongst the over 40s is less than the national average of 17.5 hours. [5] (ii) State clearly one reason why the assumption(s) you made in (i) may not be valid. [1] [Total 6](i) (ii) (iii) Sketch a graph of the cumulative distribution function for the salary dataset (both sexes combined}, assuming that all salaries lie in the range (0, 100,00} and that the)? are distributed uniformly within each range. [2] Hence estimate the median and the interquartile range for the salaries. [3] Estimate the mean salary using the same assumptions and comment on your answer. [3] [Total 3] Exercise 4.11] Calculate Am given that _ 1 _ _ am _ 0.42241, Assam _ 0.14996, am _ 0.31266. Exercise 4.11 Under an endowment insurance issued to a life aged 1', let 1' denote the present value of a unit sum insured, payable at the moment of death or at the end of the n-year term. Under a term insurance issued to a life aged 1', let 1' denote the present value of a unit sum insured, payable at the moment of death within the 11-year term. Given that 1iv'Uf]: 0.01152, it" = [1.3, \"p1 = 0.3, ELF] = calculate v'[r]. Exercise 4.12 Show that if 1:}, = log p], for y = 1,]: + 1,; + 2,..., then under the assumption of a constant force of mortality between integer ages, v {1 \"Pxi _ U1 1+! '41: tpx 5 +Ux+t Exercise 4.13 Let 2'1 denote the present value of an 11-year term insurance benet, issued to {it}. Let 22 denote the present value of a whole of life ins urance benet, issued to the same life. Express the covariance of 21 and Z3; in actuarial functions, simplied as far as possible. Exercise 4.14 You are given the following excerpt from a select life table. [I] is] It2r1+1 I[n+2 Itx1+3 6+4 1+4 [44]] losses 99399 99924 995213 99233 44 [41] 99 3'32 99 639 99 5'32 99 233 99 {133 45 [42] 99 59? 99 4?] 99 263 99 {131] 93 T52 46 [43] 99 365 99 225 99 [3}? 93 T4? 93 435 4? [44] 99 121] 93 964 93 T26 93 429 93 06? 43 Assuming an interest rate of 6% per year, calculate (a) AIMHIE ' (b) the standard deviation of the present value of a four-year term insurance, deferred one year, issued to a newly selected life aged 44], with sum insured $li, payable at the end of the year of death, and {c} the probability that the present value of the benet described in (b) is less than or equal to $35 []. Ehrercise 4.15 (a) Describe in words the insurance benets with the present values given below. i) z _ 201:1": ifr, 515, ' 1 _ tool": ifT, r:- 15. {I if T, 5 5, (ii) 2; = to 1:1": if5 a T, 5 15, to 1:15 if T, s 15. (b) Write down in integral form the formula for the expected value for (i) 2'1 and (ii) 2;. (c) Derive expressions in terms of standard actuarial functions for the expected values of 21 and 22. {d} Denve an expression in terms of standard actuarial functions for the covariance of El and 22. Exercise 4.16 Suppose that Makeham's law applies with A = BLED], H = 0.013135 and c = l.?5. Assume also that the effective rate of interest is 6% per year. (a) Use Excel and backward recursion in parts (i) and (ii). (i) Construct a table of values of A,r for integer ages, starting atx = 51}. [ii] Construct a table of values of A?\" for x = 50, 5.25, 50.5, . . .. (Do not use UDD for this.) [iii] Hence, write down the values of A 5o, Am, A23.] and Aibb- (b) Use your values for A 5;] and A 1m to estimate '51be and Aibb using the U'DD assumption. (c) Compare your estimated values for the AW functions [from [b)) with your accurate values [from [a)). Comment on the differences. Exercise 4.15r A life insurance policy issued to a life aged 5!] pays $2000 at the end of the quarter year of death before age 65 and $1{H}D at the end of the quarter year of death after age 65. Use the Standard Ultimate Survival Model, with interest at 5% per year, in the follovving. (a) Calculate the EPV of the benet. (b) Calculate the standard deviation of the present value of the benet. (c) The insurer charges a single premium of $50!}. Assuming that the insurer invests all funds at exactly 5% per year effective, what is the probability
