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From the image you uploaded, the question relates to a clinical trial that tests the effectiveness of a drug for treating insomnia. Here’s a breakdown of the problem:

### Given Information:
- **Before treatment**: Mean wake time of 25 subjects = 105.0 minutes.
- **After treatment**: Mean wake time = 84.3 minutes with a standard deviation of 20.1 minutes.
- The task asks to construct a **90% confidence interval** for the mean wake time of the population after treatment.

### Confidence Interval Calculation (already provided in the image):
- The 90% confidence interval is given as:
  \[
  77.4 \, \text{min} \leq \mu \leq 91.2 \, \text{min}
  \]
  
This means we are 90% confident that the true mean wake time for the population after the treatment falls between 77.4 and 91.2 minutes.

### Question:
It then asks whether this result suggests that the drug treatment has an effect compared to the pre-treatment wake time (105.0 minutes).

### Interpretation of the Confidence Interval:
The **mean wake time before treatment** was 105.0 minutes. Since this value is **outside** the 90% confidence interval (which ranges from 77.4 to 91.2 minutes), we can conclude that the drug treatment likely had a statistically significant effect in reducing wake time.

Thus, the answer to the final question is that **the drug treatment does have an effect**.

Would you like further details on any part of the solution? Here are five related questions you could explore:

1. How do you calculate a confidence interval manually?
2. Why is a 90% confidence interval used instead of 95% or 99%?
3. How would the confidence interval change if the sample size increased?
4. What is the meaning of standard deviation in this context?
5. What steps could be taken if the confidence interval included the pre-treatment mean?

**Tip**: Confidence intervals provide a range of plausible values for a population parameter, helping assess the effect of treatments or interventions.

A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. The mean wake time before treatment for 25 subjects was 105.0 minutes. After treatment, the mean wake time was 84.3 minutes, with a standard deviation of 20.1 minutes. Construct a 90% confidence interval estimate of the mean wake time for the population after treatment and determine whether the drug appears to be effective based on the confidence interval.

Calculate the 90% confidence interval for the mean wake time of patients after insomnia treatment. The mean wake time before treatment was 105.0 minutes, and after treatment, the mean wake time decreased to 84.3 minutes with a standard deviation of 20.1 minutes. Learn how to use the confidence interval formula to evaluate the effectiveness of the drug.

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