A simple random sample of size $n$ is drawn from a population that is normally distributed. The sample mean, $\stackrel{}{x},$ is found to be $19\mathrm{.}5,$ and the sample

standard deviation, $s,$ is found to be $4\mathrm{.}6.$

$($a$)$ Construct a $90\%$ confidence interval about $$ if the sample size, $n,$ is $36.$

$($b$)$ Construct a $90\%$ confidence interval about $$ if the sample size, $n,$ is $52.$ How does increasing the sample size affect the margin of error, E$?$

$($c$)$ Construct a $99\%$ confidence interval about $$ if the sample size, $n,$ is $36.$ How does increasing the level of confidence affect the size of the margin

of error, $E?$

$($d$)$ If the sample size is $24,$ what conditions must be satisfied to compute the confidence interval?