Question 3
$0 / 1$ pt $55 \rightleftarrows 99$
Details
If $n=28, \bar{x}(x-b a r)=38$, and $s=4$, construct a confidence interval at a $99 \%$ confidence level. Assume the data came from a normally distributed population.

Give your answers to three decimal places.
\[
< \mu<
\]

Step 1 ：Given values are \( \bar{x} = 38 \), \( s = 4 \), \( n = 28 \), and \( z = 2.576 \) for a 99% confidence level.

Step 2 ：First, calculate the margin of error using the formula \( z \times \frac{s}{\sqrt{n}} \). Substituting the given values, we get \( 2.576 \times \frac{4}{\sqrt{28}} \approx 1.947 \).

Step 3 ：Next, calculate the lower and upper bounds of the confidence interval using the formulas \( \bar{x} - \text{margin of error} \) and \( \bar{x} + \text{margin of error} \) respectively.

Step 4 ：Substituting the values, we get \( 38 - 1.947 \approx 36.053 \) for the lower bound and \( 38 + 1.947 \approx 39.947 \) for the upper bound.

Step 5 ：Thus, the 99% confidence interval for the population mean is \(\boxed{<36.053, 39.947>}\).