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 $100,$ and the sample

standard deviation, s$,$ is found to be $10.$

$($a$)$ Construct a $96\%$ confidence interval about $$ if the sample size, $n,$ is $21.$

$($b$)$ Construct a $96\%$ confidence interval about $$ if the sample size, $n,$ is $12.$

$($c$)$ Construct a $90\%$ confidence interval about $$ if the sample size, $n,$ is $21.$

$($d$)$ Should the confidence intervals in parts $($a$)-($c$)$ have been computed if the population had not been normally distributed?