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To solve this problem, we need to construct a **90% confidence interval** for the **mean wake time** after the treatment. Here is the step-by-step solution:

### Given Information:
- Sample size \(n = 25\)
- Sample mean after treatment \( \bar{x} = 84.3 \text{ minutes} \)
- Sample standard deviation after treatment \( s = 20.1 \text{ minutes} \)
- The confidence level is 90%, so the corresponding **z-score** for a 90% confidence interval is 1.645 (from standard normal distribution tables).

### Confidence Interval Formula:
The confidence interval for the mean when the population standard deviation is unknown (but the sample size is relatively small) is given by:

\[
\bar{x} \pm z \cdot \left(\frac{s}{\sqrt{n}}\right)
\]

Where:
- \( \bar{x} \) is the sample mean,
- \( z \) is the z-score for the given confidence level (1.645 for 90%),
- \( s \) is the sample standard deviation,
- \( n \) is the sample size.

### Step-by-Step Calculation:

1. **Calculate the standard error**:
   \[
   \frac{s}{\sqrt{n}} = \frac{20.1}{\sqrt{25}} = \frac{20.1}{5} = 4.02
   \]

2. **Multiply by the z-score**:
   \[
   1.645 \times 4.02 = 6.61
   \]

3. **Construct the confidence interval**:
   \[
   84.3 \pm 6.61
   \]

   So, the confidence interval is:
   \[
   (84.3 - 6.61, 84.3 + 6.61) = (77.7, 90.9) \text{ minutes}
   \]

### Final Answer:
The 90% confidence interval estimate for the mean wake time after the treatment is **77.7 min < μ < 90.9 min**.

### Interpretation:
- Since the mean wake time before the treatment was 105.0 minutes, and the confidence interval after the treatment does not include this value, it suggests that the treatment is effective in reducing the mean wake time.

---

Would you like further clarification on any part of the calculation?

### Related Questions:
1. How does the confidence interval change if the sample size increases?
2. What is the effect of increasing the confidence level from 90% to 95%?
3. Why do we use the z-score for this calculation instead of the t-score?
4. What would happen if the standard deviation were smaller?
5. How can you interpret the confidence interval in practical terms?

### Tip:
The larger the sample size, the smaller the margin of error for the confidence interval, resulting in a more precise estimate of the population mean.

A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 25 subjects had a mean wake time of 105.0 min. After treatment, the 25 subjects had a mean wake time of 84.3 min and a standard deviation of 20.1 min. Assume that the 25 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 105.0 min before the treatment? Does the drug appear to be effective?

Learn how to construct a 90% confidence interval to evaluate the effectiveness of a drug for treating insomnia in older subjects. This example provides step-by-step calculations using a sample mean wake time of 84.3 minutes and standard deviation of 20.1 minutes from 25 subjects, helping assess whether the treatment significantly reduces wake times compared to the pre-treatment mean of 105.0 minutes.

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