Question

Scenario. Bank X provides home loan service to their retail customers and charge a interest rate for their service. Due to market fluctuations, Bank X only offers floating interest rate (FIR) for their home loan customers. A friend of you who works as a banker at Bank X told you that the average yearly FIR in the last thirty years follows a Normal distribution with mean 3.2% (i.e., μ = 0.032) and standard deviation 0.04 (i.e., σ = 0.04). In addition, a sample of size 10 (drawn from N(0.032, 0.0016)) is available to you which has been summaried in the table below
-0.021 0.029 -0.009 -0.002 0.002 -0.006 0.006 -0.064 0.023 0.031 Table 1: A sample of FIR
Your friend seek help from you to use your knowledge from TSTA602 to assist Bank X to do the following data analysis. Please write a report to answer all the following questions.
(a). calculate (mannually) the sample mean, and sample standard de- viation for the sample in Table 1.
(b). if we draw samples of sizes 10 many times and form a distribution of sample mean, state the distribution of the sample mean and provide the reason for your conclusion; calculate the mean of the sample means and its standard deviation.
(c). based on (b), find the probability that the sample mean is smaller than 0.02. Please keep two decimal places in the calculation of standardization.
(d). if the sample mean from (a) is your observed value, calculate the 95% confidence interval for the sample mean.
(e). An outlier is a data point outside the interval [Q1 − 1.5IQR, Q3 + 1.5IQR], where Q1 and Q3 are first and third quartile respectively, and IQR
1

is the interquartile range. Explain whether there is any outlier appears in the sample in Table 1? You can calculate Q1 and Q3, and IQR using R (in Rstudio).
(f). Use R (in Rstudio) to generate 1,000,000 samples of size 10 from N(0.032, 0.0016) (please set the seed equals to 602), compute the sample mean for each of these samples, draw a histgram (set the frequency parameter to FALSE) for these sample means and add a density curve to the histgram. Please provide the histgram with the density curve here and attach your R code as appendix.

Answer 1

Note : Allowed to solve only 4 questions in one post.
a. Mean and standard deviation of the sample

b. The distribution of the mean will be normal.
The mean of the sample mean = 0.032
the standard deviation =

The reason for this is the central limit theroem which states that If we repeatedly take a large number of the sample of size n, then the distribution of the sample means will be normally distributed and the mean of the this distributed will be equal to the population mean. As the sample size increase, the sampling distributed tends more to the normal distribution, irrespective of the shape of the population distribution.

c. Probability of less than 0.02

d. 95% confidence interval