You may need to use the appropriate appendix table or technology to answer this question.

A simple random sample with $n=58$ provided a sample mean of $26\mathrm{.}5$ and a sample standard deviation of $4\mathrm{.}4.($Round your answers to one decimal place.$)$

$($a$)$ Develop a $90\%$ confidence interval for the population mean.

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$($b$)$ Develop a $95\%$ confidence interval for the population mean.

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$($c$)$ Develop a $99\%$ confidence interval for the population mean.

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$($d$)$ What happens to the margin of error and the confidence interval as the confidence level is increased?

As the confidence level increases, there is a smaller margin of error and a more narrow confidence interval.

As the confidence level increases, there is a smaller margin of error and a wider confidence interval.

As the confidence level increases, there is a larger margin of error and a wider confidence interval.

As the confidence level increases, there is a larger margin of error and a more narrow confidence interval.