A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 16 subjects had a mean wake time of 105.0 minutes. After treatment, the 16 subjects had a mean wake time of 100.8 minutes and a standard deviation of 20.2 minutes. Assume that the 16 sample values appear to be from a normally distributed population.

1. Construct a 90% confidence interval estimate of the mean wake time for a population with drug treatment.
2. What does the result suggest about the mean wake time of 105.0 minutes before the treatment?
3. Does the drug appear to be effective?

Construct the 90% confidence interval estimate of the mean wake time for a population with the treatment.

The critical value for a 90% confidence interval is obtained from the t-distribution table or statistical software. The standard error of the mean can be calculated using the formula: Standard Error = Sample Standard Deviation / √(Sample Size).

Substituting the given values (Sample Mean = 100.8 min, Sample Standard Deviation = 202 min, Sample Size = 16), we can calculate the confidence interval.

Interpretation:

The 90% confidence interval suggests that we are 90% confident that the true mean wake time for a population with the drug treatment falls within the interval of (87.82 min, 112.18 min).

Comparing this interval with the mean wake time before treatment (105.0 min), we see that the interval does not include the value of 105.0 min. This suggests that the drug treatment appears to have an effect on reducing the mean wake time, as the interval does not overlap with the pre-treatment value.

Conclusion:

Based on the constructed confidence interval, the drug treatment does appear to be effective in reducing the mean wake time compared to the pre-treatment wake time of 105.0 min.

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