Answer the following questions. Please explain your answers in detail.

A random sample, x. ... .$10, from a normal population gives the following values: 9.5 18.2 4.69 3.76 14.2 17.13 15.69 13.9 15.7 7.42 Ex, = 120.19 Ex, =1,693.6331 (i) Test at the 5% level whether the mean of the whole population is 15 if the variance is: (a) unknown (b) 20. [5] (ii) Test at the 5% level whether the population variance is 20. (3] [Total 8]A life office sold a portfolio of 10,000 term assurance policies on 1/1/2003. The policies had a term of 2 years, with premiums paid annually in advance and were sold to a group of males aged 60 exactly at that date. Each policy had a sum assured of $50,000, which is payable at the end of the year of death. The company prices the product assuming AM92 Ultimate mortality. (i) The same premium was charged for each year. The premium was calculated by setting the expected present value of the premiums equal to the expected present value of the benefit payments plus 10% of the standard deviation of this present value. Calculate the premium assuming 4% pa interest. [7] (Hi) During the first policy year 75 policyholders died. Calculate the net premium reserve at the end of 2003 and hence the mortality profit for the portfolio for calendar year 2003. [5] (iii) A director of the company has calculated the profit of the business as premiums received less sum assured paid on death less the net premium reserve. He calculates the profit as "just under (3.5m" and writes to ask why this conflicts with the mortality profit set out above. Show that the director's figures are numerically correct and then explain why the two figures differ. [6] [Total 18]