Construct a 90% confidence interval for the population standard deviation $\sigma$ if a sample of size 8 has standard deviation s = 20.
A) (14.11, 35.94)
B) (14.37, 34.22)
C) (15.26, 31.44)
D) (13.44, 32.01)

Question

Quizplus · Accepted Answer

To construct a 90% confidence interval for the population standard deviation $\sigma$ when the sample size (n) is 8 and the sample standard deviation (s) is 20, we use the chi-square ($\chi^2$) distribution. The formula for the confidence interval is given by $\left( \sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}} \right)$, where $n-1$ is the degrees of freedom (in this case, 7), $s^2$ is the sample variance (400), and $\chi^2_{\alpha/2}$ and $\chi^2_{1-\alpha/2}$ are the critical values from the chi-square distribution for the given confidence level (90% in this case). Using the chi-square distribution table or a calculator, the critical values for 7 degrees of freedom at a 90% confidence level are approximately 2.167 (for $\chi^2_{0.95}$) and 14.067 (for $\chi^2_{0.05}$). Plugging these values into the formula gives the confidence interval for $\sigma$ as approximately (14.11, 35.94).

Construct a 90% Confidence Interval for the Population Standard Deviation σ\sigmaσ

Construct a 90% Confidence Interval for the Population Standard Deviation $\sigma$