Start this question on a new A4 sheet. Show all necessary calculation steps. Where applicable, you should use ONLY the definitions given in your lectures, and not definitions from any external sources. All numerical answers should be rounded to TWO decimal places. For example, 0.6783 is rounded to 0.68, and 0.298 is rounded to 0.30. No rounding is needed for numerical answers of the forms, say 3 and 2.8. Question 5 (10 marks) Let X; and Y, be the annual investment returns of Superannuateon Fund A and Super- annuation Fund B in Year i, respectively, for i = 1, ...,5. Suppose you are a financial advisor and are asked by your client to analyse the performances of the two funds. The following table gives the data on the annual returns of the two funds and their differences for the years from Year 1 to Year 5. The table also provides the sample means and the sample standard deviations of the annual returns of the two funds and their differences over the 5 years. Year i X, - YA 0.070386|0.037102|0.033284 0.060112 0.048857 0.011254 0.067302 0.011983 0.055319 0.044407 0.024896 0.019511 0.066202 0.037385 0.028817 Sample Mean | 0.061682 | 0.032045 0.029637 Sample SD 0.010352 0.014057 0.016688 Based on some preliminary studies, you believe that the annual returns of each of the two funds follow a Normal distribution. Furthermore, suppose that the annual returns of the two funds are independent, and that the annual returns of each of the two funds in different years are independent. The population means ux and py of the annual returns of the two funds are unknown. Also, the population variances ox and of of the annual returns of the two funds are unknown, but they are assumed to be equal. Suppose that the population means and population variances of the annual returns of the two funds remain unchanged over these five years. Using the above given data, answer the questions in Part a) - Part e) as follows: a) Find a 95% symmetric confidence interval for the difference in the population means px - HY. (4 marks) b) Based on the result in Part a), comment on whether there is evidence to suggest a difference between the expected annual returns of the two funds. (1 mark) Now you would like to use the above given data to test whether there is a difference in the expected annual returns of the two funds. c) State the null hypothesis Ho and the alternative hypothesis H1. (1 mark) d) Determine the standardised test statistic for the data, and also clearly state its distribution under Ho- (2 marks) e) Should Ho be retained or rejected at a 5% significance level? Justify your answer. (2 marks) Disregard the box below