Questions: Determine the critical values for the confidence interval for the population standard deviation from the given values. Round your answers to three decimal places. n=7 and c=0.98

Solution Steps

To determine the confidence interval for the variance of a single population with an unknown population mean, we use the formula:

\[ \left(\frac{(n - 1)s^2}{\chi^2_{\alpha/2}}, \frac{(n - 1)s^2}{\chi^2_{1 - \alpha/2}}\right) \]

Given:

• Sample size \( n = 7 \)
• Sample variance \( s^2 = 1 \)
• Significance level \( \alpha = 0.02 \)

Substituting the values, we have:

\[ CI = \left(\frac{(7 - 1) \times 1}{\chi^2_{0.01}}, \frac{(7 - 1) \times 1}{\chi^2_{0.99}}\right) \]

Calculating the confidence interval yields:

\[ CI = (0.357, 6.88) \]

To find the confidence interval for the standard deviation, we take the square root of the variance interval:

\[ CI_{SD} = \left(\sqrt{0.357}, \sqrt{6.88}\right) \]

Calculating the square roots gives:

\[ CI_{SD} \approx (0.5975, 2.6230) \]

Final Answer