2. You are a bank manager. You want to determine the average monthly income of your clients. A random sample of 225 clients was drawn independently. The sample mean was 16450 HKD, and the sample standard deviation was 3675 HKD. (a) Find a 95%-confidence interval for the average monthly income. [20 marks] Note the following. . If d is the CMF of N(0, 1), then D(-1.96) ~ 0.025, 4(-1.64) ~ 0.05, 4(-1.28) ~ 0.01. . If you make approximations (except for the above values of ), please justify why the approximations are reasonable. (b) Is it correct that the interval you find above contains the true average monthly income with a probability of 95% (approximately)? Why? [20 marks] (c) Does the sample substantiate that the average monthly income is higher than 16000 HKD? Use a 2.5% level of significance. [20 marks]