(Solution) - A researcher did a randomization test to compare men and -(2025 Original AI-Free Solution)

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A researcher did a randomization test to compare men and women on a variable Y. The null hypothesis is that men and women are the same, and the alternative is that they are different. There are 28 possible randomizations of the data. The graph below shows the statistic "Difference in means" for each randomization. This includes the original assignment of observations to the male and female groups, which had a difference in means of 31.
(a) What is the randomization test P-value?
(b) Using ? = 0.10, is there statistically significant evidence that men and women differ with respect to the mean of variable V?
(c) The difference in sample means between men and women was 31 (mm - wome). What is the randomization test P-value for the test that considers the alternative hypothesis ?mm > ?women?