To find the critical values $\chi^2_L$ and $\chi^2_U$ for the given levels of confidence and sample sizes, we need to use the Chi-square distribution with degrees of freedom (df) calculated as $n - 1$, where $n$ is the sample size.

For each part, we'll calculate:

1. The degrees of freedom (df) as $n - 1$.
2. The lower and upper critical values corresponding to the confidence level.

Let's proceed with each part:

(a) 90% Confidence, $n = 20$

• Degrees of freedom $df = n - 1 = 20 - 1 = 19$
• For a 90% confidence interval, we split the 10% (1 - 0.90) tail probability equally on both sides, so:
  • Lower critical value ($\chi^2_L$) corresponds to the 5th percentile.
  • Upper critical value ($\chi^2_U$) corresponds to the 95th percentile.

(b) 99% Confidence, $n = 14$

• Degrees of freedom $df = n - 1 = 14 - 1 = 13$
• For a 99% confidence interval, we split the 1% (1 - 0.99) tail probability equally, so:
  • Lower critical value ($\chi^2_L$) corresponds to the 0.5th percentile.
  • Upper critical value ($\chi^2_U$) corresponds to the 99.5th percentile.

(c) 95% Confidence, $n = 25$

• Degrees of freedom $df = n - 1 = 25 - 1 = 24$
• For a 95% confidence interval, we split the 5% (1 - 0.95) tail probability equally, so:
  • Lower critical value ($\chi^2_L$) corresponds to the 2.5th percentile.
  • Upper critical value ($\chi^2_U$) corresponds to the 97.5th percentile.

I'll calculate these values.Here are the critical values for each part:

(a) 90% Confidence, $n = 20$

• Degrees of freedom $df = 19$
• $\chi^2_L = 10.117$
• $\chi^2_U = 30.144$

(b) 99% Confidence, $n = 14$

• Degrees of freedom $df = 13$
• $\chi^2_L = 3.565$
• $\chi^2_U = 29.819$

(c) 95% Confidence, $n = 25$

• Degrees of freedom $df = 24$
• $\chi^2_L = 12.401$
• $\chi^2_U = 39.364$

Would you like further explanation or have any questions?

Here are some related questions:

1. How does changing the confidence level affect the critical values?
2. What happens to the critical values when the sample size increases?
3. How are $\chi^2_L$ and $\chi^2_U$ used in constructing confidence intervals for variance?
4. Can you explain the relationship between the Chi-square distribution and degrees of freedom?
5. How would these values change if we had a one-sided confidence interval?

Tip: When using critical values, always verify the confidence level and ensure the tail probabilities are distributed correctly for two-tailed intervals.