Questions: A clinical trial was conducted to test the effectiveness of a drug used for treating insomnia in older subjects. After treatment with the drug, 29 subjects had a mean wake time of 94.4 min and a standard deviation of 41.6 min. Assume that the 29 sample values appear to be from a normally distributed population and construct a 90% confidence interval estimate of the standard deviation of the wake times for a population with the drug treatments. Does the result indicate whether the treatment is effective? Find the confidence interval estimate. min < σ < min (Round to two decimal places as needed.)

Solution Steps

The sample variance \( s^2 \) is calculated using the standard deviation provided:

\[ s^2 = (41.6)^2 = 1730.56 \]

The confidence interval for the variance of a single population with unknown population mean is given by:

\[ \left( \frac{(n - 1)s^2}{\chi^2_{\alpha/2}}, \frac{(n - 1)s^2}{\chi^2_{1 - \alpha/2}} \right) \]

Substituting the values:

\[ CI = \left( \frac{(29 - 1) \times 1730.56}{\chi^2_{0.05}}, \frac{(29 - 1) \times 1730.56}{\chi^2_{0.95}} \right) \]

Calculating the confidence interval yields:

\[ CI = (1172.21, 2862.48) \]

To find the confidence interval for the standard deviation \( \sigma \), we take the square root of the variance confidence interval:

\[ \left( \sqrt{1172.21}, \sqrt{2862.48} \right) = (34.24, 53.50) \]

Final Answer