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 $18\mathrm{.}8,$ and the sample

standard deviation, $s,$ is found to be $4\mathrm{.}7.$

$($a$)$ Construct a $90\%$ confidence interval about $$ if the sample size, $n,$ is $37.$

$($b$)$ Construct a $90\%$ confidence interval about $$ if the sample size, $n,$ is $61.$ How does increasing the sample size affect the margin of error, $E?$

$($c$)$ Construct a $95\%$ confidence interval about $$ if the sample size, $n,$ is $37.$ How does increasing the level of confidence affect the size of the

margin of error, $E?$

$($d$)$ If the sample size is $16,$ what conditions must be satisfied to compute the confidence interval?

$($a$)$ Construct a $90\%$ confidence interval about $$ if the sample size, $n,$ is $37.$

Lower bound: ; Upper bound:

$($Round to two decimal places as needed.$)$

$($b$)$ Construct a $90\%$ confidence interval about $$ if the sample size, $n,$ is $61.$

Lower bound: ; Upper bound:

$($Round to two decimal places as needed.$)$

How does increasing the sample size affect the margin of error, E$?$

A$.$ The margin of error decreases.

B$.$ The margin of error increases.

C$.$ The margin of error does not change.

$($c$)$ Construct a $95\%$ confidence interval about $$ if the sample size, $n,$ is $37.$

Lower bound: ; Upper bound:

$($Round to two decimal places as needed.$)$

Compare the results to those obtained in part $($a$).$ How does increasing the level of confidence affect the size of the margin of error, E$?$

A$.$ The margin of error does not change.

B$.$ The margin of error decreases.

C$.$ The margin of error increases.

$($d$)$ If the sample size is $16,$ what conditions must be satisfied to compute the confidence interval?

A$.$ Since the sample size is suitably large, the population need not be normally distributed, but it still should not contain any outliers.

B$.$ The sample data must come from a population that is normally distributed with no outliers.

C$.$ The sample must come from a population that is normally distributed and the sample size must be large.