Scenario. Bank X 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% (i.e., = 0.025) and variance 0.09 (i.e., = 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

Table 1: A sample of FIR

Bank X 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 X's 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 simulation task as well. You are

asked to simulate 1, 000, 000 samples for the last 10 months's 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.