Answer the following macroeconomic questions. Include explanations in your working.

Question 5. 1'4 An actuarial student nuts a stall at a children's charity fair every year. He wants to decide an appropriate fee to charge for next year. Each child is given three quoits, which they have to throw and try to \"hoop" a peg. Any child who gets three hoops over the peg wins a prize. The frequencies of successful throws last year {which the student diligeny recorded) were as follows: ---I The probability,r of any particular throw being successful can be considered to have a constant value p for all children, and the outcome of different throws can be considered to be statistically independent. (i) Derive the maximum likelihood estimate of p . [6] {ii} Find the standard error of p . [2] (iii) Find a symmetrical 90% confidence interval for p . [2] [Total 1a] Question 4.27 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 _(In P,)? = 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=at bite; where a and b are constants and the e; 's are uncorrelated N(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]