Step 1 :The problem provides the following information: the sample mean (\(\bar{x}\)) is 58 minutes, the population standard deviation (\(\sigma\)) is 8 minutes, and the sample size (\(n\)) is 25.
Step 2 :We are asked to calculate a confidence interval at the 99% confidence level. The parameters needed for this calculation are the sample mean (\(\bar{x}\)), the population standard deviation (\(\sigma\)), and the sample size (\(n\)).
Step 3 :Using these values, we can calculate the confidence interval. The formula for a confidence interval is \(\bar{x} \pm Z \frac{\sigma}{\sqrt{n}}\), where \(Z\) is the Z-score corresponding to the desired confidence level.
Step 4 :For a 99% confidence level, the Z-score is approximately 2.576.
Step 5 :Substituting the given values into the formula, we get \(58 \pm 2.576 \frac{8}{\sqrt{25}}\).
Step 6 :Solving this expression gives us the confidence interval (53.88, 62.12).