Sure! Here’s the answer in 2 steps with explanations after each step, and the final answer at the end: --- ### **Step 1: State the Null and Alternative Hypotheses** **Explanation:** To test whether the two population distributions (salaries of men and women) are identical, we set up the hypotheses as follows: - **Null hypothesis (\( H_0 \))**: The two populations of salaries are identical. - **Alternative hypothesis (\( H_a \))**: The two populations of salaries are not identical. This setup tests if there is a statistically significant difference between the salary distributions for men and women. --- ### **Step 2: Find the p-value and State the Conclusion** **Explanation:** - The test statistic \( W = 54 \). - The p-value is \( 0.0495 \). - The significance level \(\alpha = 0.05\). Since the p-value (0.0495) is **less than** the significance level (0.05), we **reject the null hypothesis**. --- ### **Final Answer:** There is sufficient evidence to conclude that there is a significant difference between the annual salaries of men and women.

Certainly! Based on the provided image, it appears to be a problem involving a Wilcoxon Rank Sum test (also known as the Mann-Whitney U test) comparing salaries of men and women. --- ### **Given Data Summary:** - **Sample size for men (n₁):** 10 - **Sample size for women (n₂):** 11 - **Sum of ranks for men (R₁):** 388 - **Sum of ranks for women (R₂):** 486 - **Test statistic (W):** 54 (from the image) - **p-value:** 0.0495 - **Significance level (α):** 0.05 --- ## **(a) State the hypotheses** **Null hypothesis (\( H_0 \))**: The distributions of salaries for men and women are identical. **Alternative hypothesis (\( H_a \))**: The distributions of salaries for men and women differ (two-sided test). --- ## **(b) Calculate the test statistic (W) and p-value** ### Step 1: Calculate the sum of ranks for men (\( R_1 \)) Given: \( R_1 = 388 \) ### Step 2: Calculate the Mann-Whitney U statistic for men (\( U_1 \)) \[ U_1 = R_1 - \frac{n_1(n_1 + 1)}{2} \] \[ U_1 = 388 - \frac{10 \times 11}{2} = 388 - 55 = 333 \] Similarly, for women (\( R_2 = 486 \)): \[ U_2 = R_2 - \frac{n_2(n_2 + 1)}{2} = 486 - \frac{11 \times 12}{2} = 486 - 66 = 420 \] Note that \( U_1 + U_2 = n_1 n_2 = 10 \times 11 = 110 \). However, the sum of the two U's should be consistent: \[ U_1 + U_2 = 333 + 420 = 753 \] But this sum exceeds the total possible \( n_1 n_2 = 110 \), indicating the ranks are totaled differently, or the W value provided (54) is the actual test statistic used for the p-value. --- ### Step 3: Use the provided test statistic \( W = 54 \) According to the image, the test statistic is \( W = 54 \), and the p-value associated with this is 0.0495. **Therefore**, the p-value from the Wilcoxon rank-sum test is **0.0495**. --- ## **Conclusion:** Since the p-value **(0.0495)** is **less than** the significance level \( \alpha = 0.05 \), **we reject the null hypothesis**. --- ## **Final answers:** ### **(a)** - \( H_0 \): Salaries of men and women come from the same distribution. - \( H_a \): Salaries of men and women do not come from the same distribution. ### **(b)** - The test statistic \( W = 54 \). - The p-value = **0.0495**. - **Decision**: Reject \( H_0 \); conclude there is a statistically significant difference in salaries between men and women at the 5% significance level. --- **Summary:** There is sufficient evidence to suggest that the salary distributions for men and women differ significantly.