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 104.0 minutes. After treatment, the 19 subjects had a mean wake time of 100.2 minutes with a standard deviation of 20.3 minutes. Assume that the 19 sample values appear to be from a normally distributed population.

Construct a [tex]$90\%$[/tex] confidence interval estimate of the mean wake time for a population with drug treatments.

[tex]\[\square \, \text{min} \ \textless \ \mu \ \textless \ \square \, \text{min}\][/tex]

(Round to one decimal place as needed.)

What does the result suggest about the mean wake time of 104.0 minutes before the treatment? Does the drug appear to be effective?

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

### Step 1: Identify the Given Information
- Sample Size ([tex]$ n $[/tex]): 19 subjects
- Sample Mean after Treatment ([tex]$ \bar{x} $[/tex]): 100.2 minutes
- Sample Standard Deviation ([tex]$ s $[/tex]): 20.3 minutes
- Confidence Level: 90%

### Step 2: Calculate the T-Score
Since the sample size is small ([tex]$ n = 19 $[/tex]), we use the t-distribution to calculate the confidence interval. We need the t-score for a 90% confidence level with [tex]$ n-1 = 18 $[/tex] degrees of freedom.

### Step 3: Calculate the Standard Error
The standard error (SE) of the mean is calculated using the formula:
[tex]\[
SE = \frac{s}{\sqrt{n}} = \frac{20.3}{\sqrt{19}}
\][/tex]

### Step 4: Calculate the Margin of Error
The margin of error (ME) is given by:
[tex]\[
ME = t \times SE
\][/tex]
Where [tex]$ t $[/tex] is the t-score calculated earlier.

### Step 5: Construct the Confidence Interval
The confidence interval for the population mean is then:
[tex]\[
\bar{x} \pm ME
\][/tex]
This gives us the lower and upper bounds of the confidence interval:
- Lower Bound: 100.2 - ME
- Upper Bound: 100.2 + ME

### Result
After performing these calculations, the 90% confidence interval is approximately:
- Lower Bound: 92.1 minutes
- Upper Bound: 108.3 minutes

### Interpretation
This confidence interval suggests that with 90% confidence, the true mean wake time after treatment is between 92.1 minutes and 108.3 minutes. Comparing this interval to the mean wake time of 104.0 minutes before treatment:

- Since the entire interval includes 104.0 minutes, it's possible the mean wake time has not significantly changed.
- Therefore, this result does not provide strong evidence that the drug is effective in reducing wake times, as 104.0 minutes falls within the range of the confidence interval.