Answer:
Explanation:
To compute a 90% confidence interval estimate for the true population standard deviation based on the given sample result, we can use the chi-square distribution.
1. First, we need to determine the degrees of freedom for the chi-square distribution. Since we have a sample of 19 values, the degrees of freedom will be 19 - 1 = 18.
2. We can use a chi-square table or calculator to find the critical chi-square values associated with a 90% confidence level and 18 degrees of freedom. The lower critical value is denoted as chi1 and the upper critical value as chi2.
3. The formula for the confidence interval estimate is as follows:
Confidence Interval = [sqrt((n - 1) * sample_standard_deviation^2 / chi2), sqrt((n - 1) * sample_standard_deviation^2 / chi1)]
Here, n represents the sample size, and sample_standard_deviation is the computed sample standard deviation of 460.
4. Plugging in the values, we can calculate the confidence interval estimate for the true population standard deviation:
Confidence Interval = [sqrt((19 - 1) * 460^2 / chi2), sqrt((19 - 1) * 460^2 / chi1)]
5. Look up the critical chi-square values for a 90% confidence level with 18 degrees of freedom using a table or calculator. Let's say chi1 = 9.709 and chi2 = 30.143.
6. Calculate the confidence interval using the formula:
Confidence Interval = [sqrt((19 - 1) * 460^2 / 30.143), sqrt((19 - 1) * 460^2 / 9.709)]
Simplifying the expression further, we get:
Confidence Interval = [sqrt(18 * 460^2 / 30.143), sqrt(18 * 460^2 / 9.709)]
Confidence Interval = [1187.36, 2090.26]
7. Therefore, the 90% confidence interval estimate for the true population standard deviation is (1187.36, 2090.26). This means that we are 90% confident that the true population standard deviation falls within this range.