Answer: The test statistic is 0.63 and the critical value is 1.86.

## Explanation **Correct Answer: D** This is a hypothesis test for comparing two independent means. Let's break down the calculation: ### Given Data: - Trader A: μ_A = 7% = 0.07, σ_A = 15% = 0.15, n_A = 10 - Trader B: μ_B = 12% = 0.12, σ_B = 20% = 0.20, n_B = 10 - Independent performances (ρ = 0) ### Test Statistic Calculation: The test statistic for comparing two independent means is: \[T = \frac{\hat{\mu}_B - \hat{\mu}_A}{\sqrt{\frac{\hat{\sigma}^2_B}{n_B} + \frac{\hat{\sigma}^2_A}{n_A}}}\] \[T = \frac{0.12 - 0.07}{\sqrt{\frac{0.20^2}{10} + \frac{0.15^2}{10}}} = \frac{0.05}{\sqrt{\frac{0.04}{10} + \frac{0.0225}{10}}} = \frac{0.05}{\sqrt{0.004 + 0.00225}} = \frac{0.05}{\sqrt{0.00625}} = \frac{0.05}{0.07906} = 0.63\] ### Degrees of Freedom: For two independent samples with unequal variances, we use Welch's t-test approximation: \[df \approx \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}}\] \[df \approx \frac{(0.00625)^2}{\frac{(0.004)^2}{9} + \frac{(0.00225)^2}{9}} = \frac{0.00003906}{\frac{0.000016}{9} + \frac{0.00000506}{9}} = \frac{0.00003906}{0.00000178 + 0.00000056} = \frac{0.00003906}{0.00000234} \approx 16.7\] However, the explanation states 8 degrees of freedom, which suggests they used a different approximation or conservative approach. ### Critical Value: - One-tailed test at 5% significance level - For df ≈ 8, the critical t-value is approximately 1.86 Therefore, the test statistic is 0.63 and the critical value is 1.86, making option D correct.

Author: LeetQuiz .