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 $115,$ and the sample

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

$($a$)$ Construct a $96\%$ confidence interval about $$ if the sample size, $n,$ is $28.$

$($b$)$ Construct a $96\%$ confidence interval about $$ if the sample size, $n,$ is $14.$

$($c$)$ Construct an $80\%$ confidence interval about $$ if the sample size, $n,$ is $28.$

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

$EE$ Click the icon to view the table of areas under the t$-$distribution.

$($a$)$ Construct a $96\%$ confidence interval about $$ if the sample size, $n,$ is $28.$

Lower bound: ; Upper bound:

$($Use ascending order. Round to one decimal place as needed.$)$

$($b$)$ Construct a $96\%$ confidence interval about $$ if the sample size, $n,$ is $14.$

Lower bound: ; Upper bound:

$($Use ascending order. Round to one decimal place as needed.$)$

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

A$.$ As the sample size decreases, the margin of error stays the same.

B$.$ As the sample size decreases, the margin of error increases.

C$.$ As the sample size decreases, the margin of error decreases.

$($c$)$ Construct an $80\%$ confidence interval about $$ if the sample size, $n,$ is $28.$

Lower bound: ; Upper bound:

$($Use ascending order. Round to one decimal place as needed.$)$

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

A$.$ As the level of confidence decreases, the size of the interval stays the same.

B$.$ As the level of confidence decreases, the size of the interval increases.

C$.$ As the level of confidence decreases, the size of the interval decreases.

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

A$.$ Yes, the population does not need to be normally distributed.

B$.$ No$,$ the population does not need to be normally distributed.

C$.$ Yes, the population needs to be normally distributed.

D$.$ No$,$ the population needs to be normally distributed.