Answer :
Final answer:
We can construct a 90% confidence interval for the standard deviation of the wake times using Chi-square distribution. This interval will give the range in which the true standard deviation lies with 90% confidence, but cannot directly provide evidence about the effectiveness of the insomnia treatment.
Explanation:
This question pertains to the concept of constructing confidence intervals, specifically a 90% confidence interval for the standard deviation of the wake times. To solve the problem, we will use Chi-square distribution because the standard deviation of a population is estimated here.
We can use the following formulas to find the confidence interval:
Lower bound = sqrt((n-1)s^2 / X_(1-α/2, n-1)), Upper bound = sqrt((n-1)s^2 / X_(α/2, n-1))
Where n is the sample size, s is the sample standard deviation, and X is the Chi-square value.
After substituting the values of n (16), s (42.9), and Chi-square values for (α/2, n-1) and (1-α/2, n-1) from a Chi-square table( for df=n-1=15), we would get the confidence interval. However, this interval will only give us an idea about the variability of the wake times, it will not directly indicate the effectiveness of the treatment. To infer about the treatment’s efficacy based on this confidence interval, additional information like comparison between wake times before and after the treatment might be required.
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