1. **Problem Statement:** Find the critical values for the given confidence intervals without calculating the intervals themselves. 2. **Part A: 90% confidence interval with $t$-distribution and 37 degrees of freedom.** - Formula: $\bar{X} \pm t_{\alpha/2} \left( \frac{S}{\sqrt{n}} \right)$ - Confidence level: 90% means $\alpha = 1 - 0.90 = 0.10$. - Since it's two-tailed, $\alpha/2 = 0.05$. - Degrees of freedom (df) = 37. - The critical value $t_{\alpha/2}$ is the $t$-score with 37 df such that the area to the right is 0.05. 3. **Part B: 98% confidence interval for variance using chi-square distribution with 24 degrees of freedom.** - Formula: $(n - 1) \frac{S^2}{\chi^2_R} \leq \sigma^2 < (n - 1) \frac{S^2}{\chi^2_L}$ - Confidence level: 98% means $\alpha = 1 - 0.98 = 0.02$. - Two-tailed, so $\alpha/2 = 0.01$. - Degrees of freedom (df) = 24. - $\chi^2_L$ is the chi-square value with 24 df such that the area to the left is $\alpha/2 = 0.01$. - $\chi^2_R$ is the chi-square value with 24 df such that the area to the right is $\alpha/2 = 0.01$. 4. **Part C: 94% confidence interval for proportion using normal distribution.** - Formula: $\hat{p} \pm Z_{\alpha/2} \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }$ - Confidence level: 94% means $\alpha = 1 - 0.94 = 0.06$. - Two-tailed, so $\alpha/2 = 0.03$. - The critical value $Z_{\alpha/2}$ is the $Z$-score such that the area to the right is 0.03. 5. **Summary of critical values:** - $t_{0.05, 37}$: Use $t$-distribution table or calculator to find the value with 37 df and 0.05 in the upper tail. - $\chi^2_L = \chi^2_{0.01, 24}$: chi-square value with 24 df and 0.01 area to the left. - $\chi^2_R = \chi^2_{0.99, 24}$: chi-square value with 24 df and 0.01 area to the right (equivalently 0.99 to the left). - $Z_{0.03}$: standard normal value with 0.03 area to the right. 6. **Approximate values from standard tables:** - $t_{0.05, 37} \approx 1.687$ - $\chi^2_{0.01, 24} \approx 11.688$ - $\chi^2_{0.99, 24} \approx 42.980$ - $Z_{0.03} \approx 1.880$ These are the critical values you would use for the confidence intervals as requested.