To construct the confidence interval for the population variance $\sigma^2$ and standard deviation $\sigma$ based on the provided sample variance $s^2 = 12.25$, sample size $n = 25$, and confidence level $c = 0.99$, we will use the Chi-squared distribution.

Steps to Calculate the Confidence Interval for Variance

1. Determine the Degrees of Freedom: $df = n - 1 = 25 - 1 = 24$

2. Find the Chi-squared critical values:

   • For a 99% confidence level, the significance level $\alpha = 1 - 0.99 = 0.01$.
   • The critical values are found using the Chi-squared distribution:
     • $\chi^2_{\alpha/2}$ for $df = 24$
     • $\chi^2_{1 - \alpha/2}$ for $df = 24$

   Using a Chi-squared table or calculator:

   • $\chi^2_{0.005, 24} \approx 42.98$
   • $\chi^2_{0.995, 24} \approx 9.59$

3. Calculate the Confidence Interval for Variance: $\left( \frac{(n-1)s^2}{\chi^2_{1 - \alpha/2}}, \frac{(n-1)s^2}{\chi^2_{\alpha/2}} \right)$

   Plugging in the values: $\left( \frac{24 \times 12.25}{42.98}, \frac{24 \times 12.25}{9.59} \right)$

   Calculating the limits:

   • Lower limit: $\frac{24 \times 12.25}{42.98} \approx 6.87$
   • Upper limit: $\frac{24 \times 12.25}{9.59} \approx 30.66$

Final Confidence Interval for Variance

$\text{Confidence interval for } \sigma^2: (6.87, 30.66)$

Confidence Interval for Standard Deviation

To find the confidence interval for the population standard deviation $\sigma$, simply take the square roots of the variance limits: $\left( \sqrt{6.87}, \sqrt{30.66} \right) \approx (2.62, 5.53)$

Final Confidence Interval for Standard Deviation

$\text{Confidence interval for } \sigma: (2.62, 5.53)$

Would you like more details or have any questions about this process? Here are some related questions for further exploration:

1. How does the choice of confidence level affect the width of the confidence interval?
2. What assumptions must be met for using the Chi-squared distribution in this context?
3. How can we interpret the confidence intervals for variance and standard deviation?
4. What are the implications of a larger sample size on the confidence intervals?
5. How does the method for calculating confidence intervals differ for non-normal distributions?

Tip: Always check if the underlying assumptions of your statistical methods are met before interpreting results.