Question 2 [20 marks] a) An insurance company reported that the average yearly income of their sales managers in the year 2012 was 110 ($'000). A sample of 18 sales managers, which was taken in 2013, showed an average yearly income of 105 ($'000) with a standard deviation of 36 ($'000). Test whether there has been a significant decrease in the average yearly income of sales managers in the year 2013. Assuming the yearly income is normally distributed, answer the following questions: i. State the appropriate null and alternative hypotheses. [2 marks] ii. Use a diagram to set up the rejection region and state the critical value at 1% significance level. [2 marks] D2QBM117 (August 2021 Semester, Online Examination) Page 3 iii. Outline the decision rule. [1 mark] iv. Calculate the value of the test statistic. [3 marks] v. Make a decision and conclusion. [2 marks] vi. Is there a significant decrease in the average yearly income if the value of the test statistics is -3.2? Why? [2 marks] b) An operator receives an average of 5 calls per hour. Find the probability that she will receive: i. Exactly three calls in 30 minutes. [3 marks] ii. No call in one hour. [2 marks] iii. Find the mean and standard deviation of the number of calls receive in an hour. [3 marks]