A sociologist develops a test to measure attitudes about public transportation, and 27 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.4. Construct the 95% confidence interval for the standard deviation, ?, of the scores of all subjects.
A) $17.2

Question

A sociologist develops a test to measure attitudes about public transportation, and 27 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.4. Construct the 95% confidence interval for the standard deviation, ?, of the scores of all subjects. 
A) $17.2 < \sigma < 27.2$
B) $16.9 < \sigma < 29.3$
C) $16.6 < \sigma < 28.6$
D) $17.5 < \sigma < 27.8$

Quizplus · Accepted Answer

To construct a 95% confidence interval for the standard deviation, we use the chi-square distribution because the standard deviation is related to the variance, and the variance follows a chi-square distribution. The formula for the confidence interval of the standard deviation is given by: $\sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2}}}$ and $\sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}}$, where $n$ is the sample size, $s$ is the sample standard deviation, and $\chi^2_{\alpha/2}$ and $\chi^2_{1-\alpha/2}$ are the chi-square values for $\alpha/2$ and $1-\alpha/2$, respectively, with $n-1$ degrees of freedom. Given $n=27$, $s=21.4$, and a 95% confidence level ($\alpha=0.05$), we find the chi-square values for 26 degrees of freedom and calculate the interval. The correct interval, based on the chi-square distribution and the given data, is $16.9 < \sigma < 29.3$, which corresponds to option B.

A Sociologist Develops a Test to Measure Attitudes About Public 17.2<σ<27.217.2 < \sigma < 27.217.2<σ<27.2

A Sociologist Develops a Test to Measure Attitudes About Public $17.2 < \sigma < 27.2$