Refer to the following attachments and address the questions effectively. Include all relevant information in your working.

A temporary annuity of $3,000 pa payable quarterly in arrears for a term of 10 years was purchased one year ago by Jim, on his 60th birthday, by payment of a single premium. Show algebraically that the current retrospective and prospective net reserves are equal assuming that the premium and reserving bases are the same. Ignore expenses. [4]The government of a country suffering from hyperinflation has sponsored an economist to monitor the price of a "basket" of items in the population's staple diet over a one-year period. As part of his study, the economist selected six days during the year and on each of these days visited a single nightclub, where he recorded the price of a pint of lager. His report showed the following prices: Day (i) 8 29 57 92 141 148 Price ( P. ) 15 17 22 51 88 95 In P 2.7081 2.8332 3.0910 3.9318 4.4773 4.5539 [i=475 i =54,403 [InP, = 21.5953 (InP;) =81.1584 Liln P, = 1,947.020 The economist believes that the price of a pint of lager in a given bar on day i can be modelled by: In P = a + bite; where a and b are constants and the e; 's are uncorrelated /(0,o-) random variables. (i) Estimate a, b and o? . [5] (ii) Calculate the linear correlation coefficient r. [1] (iii) Obtain a 99% confidence interval for b . [2] (iv) Determine a 95% confidence interval for the average price of a pint of lager on day 365: (a) in the country as a whole (b ) in a randomly selected bar. [7] [Total 15]