We are asked to construct a 95% confidence interval for the mean wake time after the treatment and assess whether the drug treatment appears effective in reducing wake time.

Step 1: Given information

• Sample size $n = 18$
• Sample mean after treatment $\bar{x} = 81.3$ minutes
• Sample standard deviation $s = 21.6$ minutes
• Confidence level $95\%$

Step 2: Calculate the margin of error (E)

The formula for the margin of error for a confidence interval of the mean is:

$E = t_{\alpha/2} \times \frac{s}{\sqrt{n}}$

where:

• $t_{\alpha/2}$ is the critical value of the t-distribution for $\alpha = 0.05$ (for a 95% confidence level) with $df = n - 1 = 18 - 1 = 17$.
• $s$ is the sample standard deviation.
• $n$ is the sample size.

Using a t-table or calculator, the critical value $t_{0.025, 17}$ for a 95% confidence level with 17 degrees of freedom is approximately 2.110.

Now, calculate the margin of error:

$E = 2.110 \times \frac{21.6}{\sqrt{18}} = 2.110 \times \frac{21.6}{4.2426} \approx 10.75$

Step 3: Confidence interval

The confidence interval is given by:

$\bar{x} - E < \mu < \bar{x} + E$

Substituting the values:

$81.3 - 10.75 < \mu < 81.3 + 10.75$

$70.5 < \mu < 92.1$

Step 4: Interpretation

The 95% confidence interval for the mean wake time after treatment is between 70.5 minutes and 92.1 minutes.

Step 5: Effectiveness of the drug

The mean wake time before the treatment was 103.0 minutes. Since the entire confidence interval (70.5 min to 92.1 min) lies below the pre-treatment mean of 103.0 minutes, we can conclude that the drug treatment does appear to be effective in reducing wake time in older subjects.

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Would you like any further details or have any questions? Here are some related questions to deepen your understanding:

1. How is the t-distribution different from the normal distribution in this context?
2. Why is the t-distribution used instead of the z-distribution here?
3. What would the confidence interval look like if we had a larger sample size?
4. How does the confidence level affect the width of the confidence interval?
5. Can we conclude causality from this confidence interval analysis alone?

Tip: A larger sample size reduces the margin of error, leading to a narrower confidence interval.