A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 18 subjects had a mean wake time of 102.0 min. After treatment, the 18 subjects had a mean wake time of 74.6 min and a standard deviation of 20.9 min. Assume that the 18 sample values appear to be a normally distributed population and construct a 99% confidence interval estimate of the mean wake time for a population with drug treatments. What does the result suggest about the mean wake time of 102.0 min before the treatment? Does the drug appear to be effective? Construct the $99 \%$ confidence interval estimate of the mean wake time for a population with the treatment. $\min <\mu<\square \min$ (Round to one decimal place as needed.)

Step 1 ：We are given that the sample mean wake time after treatment, denoted as \(\bar{x}\), is 74.6 minutes. The standard deviation of the sample, denoted as \(\sigma\), is 20.9 minutes. The sample size, denoted as \(n\), is 18. We are asked to construct a 99% confidence interval for the mean wake time after treatment.

Step 2 ：The formula for a confidence interval is \(\bar{x} \pm Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}\), where \(Z_{\alpha/2}\) is the Z-score corresponding to the desired confidence level. For a 99% confidence level, \(Z_{\alpha/2}\) is approximately 2.576.

Step 3 ：Substituting the given values into the formula, we get \(74.6 \pm 2.576 \cdot \frac{20.9}{\sqrt{18}}\).

Step 4 ：Calculating the margin of error, we get approximately 12.7 minutes.

Step 5 ：Subtracting and adding the margin of error from the sample mean, we get the 99% confidence interval for the mean wake time after treatment as approximately (61.9 minutes, 87.3 minutes).

Step 6 ：This means that we are 99% confident that the true mean wake time after treatment for the population is between 61.9 minutes and 87.3 minutes.

Step 7 ：\(\boxed{61.9 \min <\mu<87.3 \min}\) is the final answer.