Scenario. Bank H provides home loan service to their retail customers and charge an interest rate for their service.The interest rate fluctuates from months to months as it depends on some market factors. This rate is referred to as floating interest rate (FIR). The historical monthly FIR follows a Normal distribution with mean 2.5%, ( = 0.025) and variance 0.09, ( = 0.3). Over the last 10 months the monthly FIRs are:

-0.372, 0.006, -0.280, -0.227, -0.200, -0.261, -0.174, -0.696, -0.044, 0.014.

Bank H believes that the average of FIRs over the next year also follows a Normal distribution with the mean and the variance equal to the mean and the variance of the above period correspondingly. The prediction by one of the Bank Xs analyst is that there is more than 40% of possibility that the average of FIRs over the next year is less than 2.2%. Your first task, as a business analyst of Bank X, is to verify the claim of your collegue (without assistance from any software). Morever, your supervisor would like you to complete a simulation task as well. You are asked to simulate 1, 000, 000 samples for the last 10 monthss FIRs and construct a histogram for these sample means by using the statistical software R. Please write a short report to complete these two tasks.

For the first task, please include all the detailed calculations.

For the second task, please include an explaination on how you generate the histogram and with your R code as attachment.