A researcher collected sample data for $20$ middle$-$aged women. The sample had a mean serum cholesterol level $($measured in milligrams per one hundred milliliters$)$ of $192\mathrm{.}2,$ with a standard deviation of $5\mathrm{.}9.$ Assuming that serum cholesterol levels for middle$-$aged women are normally distributed, find a $95\%$ confidence interval for the mean serum cholesterol level of all women in this age group. Give the lower limit and upper limit of the $95\%$ confidence interval.

Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. $($If necessary, consult a list of formulas.$)$