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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 98.1 minutes and a standard deviation of 44.9 minutes. 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?

1. Find the confidence interval estimate.

2. Does the result indicate whether the treatment is effective?

A. Yes, the confidence interval indicates that the treatment is not effective because the interval contains 0 minutes.

B. Yes, the confidence interval indicates that the treatment is effective because the interval contains 0 minutes.

C. Yes, the confidence interval indicates that the treatment is effective because the interval does not contain 0 minutes.

D. No, the confidence interval does not indicate whether the treatment is effective.

E. Yes, the confidence interval indicates that the treatment is not effective because the interval does not contain 98.1 minutes.

F. Yes, the confidence interval indicates that the treatment is effective because the interval does not contain 98.1 minutes.

G. Yes, the confidence interval indicates that the treatment is not effective because the interval does not contain 0 minutes.

Answer :

The 90% confidence interval estimate is (36.93, 62.39) minutes. The confidence interval does not indicate whether the treatment is effective. Therefore, the correct option is D.

To find the 90% confidence interval estimate for the standard deviation of the wake times, we will follow these steps:

1. Identify the sample size (n), sample standard deviation (s), and degrees of freedom (df):

n = 29, s = 44.9 min, and df = n - 1 = 28.
2. Look up the chi-square values for a 90% confidence interval:

For df = 28, the chi-square values are 15.51 (lower) and 42.98 (upper).

3. Calculate the confidence interval estimate:

Lower limit = √((n - 1) * s^2 / upper chi-square value) = √(28 * (44.9)^2 / 42.98) = 36.93 min

Upper limit = √((n - 1) * s^2 / lower chi-square value) = √(28 * (44.9)^2 / 15.51) = 62.39 min

The 90% confidence interval estimate for the standard deviation of wake times is (36.93, 62.39) minutes.

As for the effectiveness of the treatment, the confidence interval does not indicate whether the treatment is effective or not, because the interval is about the standard deviation of the wake times, not the mean wake times or the success of the treatment. Therefore, the answer is D. No, the confidence interval does not indicate whether the treatment is effective.

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