$5\mathrm{.}6$ Consider a population that has a normal distribution with mean $36$ and standard deviation $8.$

$($a$)$ The sampling distribution of for samples of size $200$ will have what mean, standard error, and shape?

$($b$)$ Use R to draw a random sample of size $200$ from this population. Conduct EDA on your sample.

$($c$)$ Compute the bootstrap distribution for your sample and note the bootstrap mean and standard error.

$($d$)$ Compare the bootstrap distribution to the theoretical sampling distribution by creating a table like Table $5\mathrm{.}1.$

$($e$)$ Repeat for sample sizes of n$=50$ and n$=10.$ Carefully describe your observations about the effects of sample size on the bootstrap distribution.