Read through the attachments below and answer the questions therein. Kindly address all questions.

On a particular run of a process which bottles a drink, it is thought that the cleansing process of the bottles has partially failed. The bottles have been boxed into crates, each containing six bottles. It is thought that each bottle, independently of all others, has the same chance of containing impurities. A survey has been conducted, and each bottle in a random sample of 200 crates has been tested for impurities. The table below gives the numbers of crates in the sample which had the respective number of bottles which contained impurities. Number of impure bottles: 0 2 5 Number of crates: 38 70 58 25 6 Test the goodness of fit of a binomial distribution to these observations.Calculate numerical values for the following actuarial functions assuming the interest rates shown in brackets: (i) an (71/2% pa) (ii) (la )a (10% pa) (ii) & (4) 151 (10% pa convertible quarterly) (iv) ('%) 10 (1% pa) (v) a(4) 14% (4% pa) (vi) 514 :5(6% pa)The table below shows the progress of a lottery winner's investment portfolio for the 2006 calendar year: Fund value Cashflow I January $75,000 31 March $90,000 I April Won Lottery. Winnings invested: (2,700,000 30 September $2,600,000 1 October Funds withdrawn to pay off gambling debt: (2,500,000 31 December $125,000 Calculate (to the nearest %) the MWRR and the TWRR for 2006 for this portfolio and comment on your answers