Suppose monthly starting salaries for new college graduates in economics follow a normal

distribution with mean of $$5500$ and standard deviation of $$2000$ and you are interested

in estimating the population mean of $$5500$ based on a survey of $n$ graduates. Conduct the

following simulations to see the sampling distribution of $\stackrel{}{x}={\displaystyle \underset{1}{\overset{n}{}}}\frac{x_i}{n},$ where $x_i$ is a monthly

starting salary of $i$ th surveyee.

$($a$)$ Generate $n=10$ data following $N(5500,{2000}^2)$ and calculate the sample average $\stackrel{}{x}.$

Repeat $100$ times and draw a histogram of $\stackrel{}{x}$ s$,$ using using a class width of $$200$ with

the first class at $$4000$ and the last class at $$8000.$

$($b$)$ Repeat $($a$)$ with $n=100$ data and compare the results.

$($c$)$ Compare the sample average of ${\stackrel{}{x}}_s$ generated in $($a$)$ and $($b$).$ What is the expected

difference? Why?