Question

A clinical trial was conducted to test the effectiveness of a drug used for treating insomnia in older subjects. After treatment with the​ drug, 23 subjects had a mean wake time of 95.8 min and a standard deviation of 41.4 min. Assume that the 23 sample values appear to be from a normally distributed population and construct a ​98% 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?

Answer 1

To construct a 95% confidence interval for the mean number of hours slept for the population, we first calculate the sample mean and standard error. Then, using the t-value, we construct the confidence interval. The interval is (8.0851, 9.2315).

Step-by-step explanation:

To construct a confidence interval for the mean number of hours slept for the population, we can use the formula:

Confidence Interval = sample mean +/- (t-value * standard error)

First, we calculate the sample mean of the hours slept by summing all the values and dividing by the sample size: (8.2 + 9.1 + 7.7 + 8.6 + 6.9 + 11.2 + 10.1 + 9.9 + 8.9 + 9.2 + 7.5 + 10.5) / 12 = 103.9 / 12 = 8.6583

Next, we calculate the standard error, which is the standard deviation of the sample divided by the square root of the sample size: 0.9023 / sqrt(12) = 0.2604

Finally, we calculate the t-value using a t-table or a calculator. With a 95% confidence level and a sample size of 12, the t-value is approximately 2.201

Using these values, we can construct the confidence interval: 8.6583 +/- (2.201 * 0.2604) = (8.6583 - 0.5732, 8.6583 + 0.5732) = (8.0851, 9.2315)

Therefore, the 95% confidence interval for the mean number of hours slept for the population is (8.0851, 9.2315).