Question

construct the indicated confidence intervals for the population variance (sigma^{2}) and the population standard deviation (sigma). assume the sample is from a normally distributed population.
(c = 0.95,s^{2}=15.10,n = 25)
o a. confidence interval for (sigma^{2}): (9.62, 24.80); confidence interval for (sigma): (3.10, 4.98)
o b. confidence interval for (sigma^{2}): (9.21, 29.22); confidence interval for (sigma): (3.03, 5.41)
o c. confidence interval for (sigma^{2}): (13.72, 26.17); confidence interval for (sigma): (3.70, 5.12)
o d. confidence interval for (sigma^{2}): (8.92, 27.62); confidence interval for (sigma): (2.99, 5.26)

Explanation:

Step1: Find the degrees of freedom

The degrees of freedom \( df=n - 1\). Given \(n = 25\), then \(df=25-1 = 24\).

Step2: Find the critical values

For a confidence level \(c = 0.95\), the significance level \(\alpha=1 - c=1 - 0.95 = 0.05\). The critical values are \(\chi_{1-\frac{\alpha}{2}}^{2}\) and \(\chi_{\frac{\alpha}{2}}^{2}\).
\(\frac{\alpha}{2}=\frac{0.05}{2}=0.025\) and \(1-\frac{\alpha}{2}=1 - 0.025 = 0.975\).
From the \(\chi^{2}\) - distribution table, \(\chi_{0.975,24}^{2}=12.401\) and \(\chi_{0.025,24}^{2}=39.364\).

Step3: Construct the confidence interval for \(\sigma^{2}\)

The formula for the confidence interval for \(\sigma^{2}\) is \(\frac{(n - 1)s^{2}}{\chi_{\frac{\alpha}{2}}^{2}}\leq\sigma^{2}\leq\frac{(n - 1)s^{2}}{\chi_{1-\frac{\alpha}{2}}^{2}}\).
Substitute \(n = 25\), \(s^{2}=15.10\), \(\chi_{0.025,24}^{2}=39.364\) and \(\chi_{0.975,24}^{2}=12.401\) into the formula.
\(\frac{(25 - 1)\times15.10}{39.364}\leq\sigma^{2}\leq\frac{(25 - 1)\times15.10}{12.401}\)
\(\frac{24\times15.10}{39.364}\leq\sigma^{2}\leq\frac{24\times15.10}{12.401}\)
\(\frac{362.4}{39.364}\leq\sigma^{2}\leq\frac{362.4}{12.401}\)
\(9.21\leq\sigma^{2}\leq29.22\)

Step4: Construct the confidence interval for \(\sigma\)

Take the square - root of the endpoints of the confidence interval for \(\sigma^{2}\).
\(\sqrt{9.21}\leq\sigma\leq\sqrt{29.22}\)
\(3.03\leq\sigma\leq5.41\)

Answer:

B. Confidence interval for \(\sigma^{2}\): \((9.21,29.22)\); Confidence interval for \(\sigma\): \((3.03,5.41)\)