A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 19 subjects had a mean wake time of $101.0 \mathrm{~min}$. After treatment, the 19 subjects had a mean wake time of $79.1 \mathrm{~min}$ and a standard deviation of $20.5 \mathrm{~min}$. Assume that the 19 sample values appear to be from a normally distributed population and construct a $90 \%$ 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 $101.0 \mathrm{~min}$ before the treatment? Does the drug appear to be effective? Construct the $90 \%$ 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.) What does the result suggest about the mean wake time of $101.0 \mathrm{~min}$ before the treatment? Does the drug appear to be effective? The confidence interval the mean wake time of 101.0 min before the treatment, so the means before and after the treatment This result suggests that the drug treatment an effect.

Step 1 ：Given that the sample size (n) is 19, the mean wake time after treatment (mean_after) is 79.1 minutes, and the standard deviation (std_dev) is 20.5 minutes.

Step 2 ：We are asked to construct a 90% confidence interval for the mean wake time after treatment. The z-score for a 90% confidence level is approximately 1.645.

Step 3 ：The formula for a confidence interval is \(\bar{x} \pm z \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, z is the z-score corresponding to the desired confidence level, s is the sample standard deviation, and n is the sample size.

Step 4 ：Substituting the given values into the formula, we get \(79.1 \pm 1.645 \frac{20.5}{\sqrt{19}}\).

Step 5 ：Calculating the margin of error gives approximately 7.7364720854500195 minutes.

Step 6 ：Subtracting and adding this margin of error from the mean wake time after treatment gives the confidence interval (71.36352791454998, 86.83647208545001).

Step 7 ：Rounding to one decimal place, the confidence interval is approximately (71.4 min, 86.8 min).

Step 8 ：Since this interval does not include the mean wake time before treatment (101.0 min), this suggests that the drug is effective in reducing wake time.

Step 9 ：\(\boxed{\text{The 90% confidence interval estimate of the mean wake time for a population with the treatment is 71.4 min < \mu < 86.8 min. This result suggests that the drug treatment has an effect in reducing the mean wake time from 101.0 min before the treatment. Therefore, the drug appears to be effective.}}\)