The Mann-Whitney U Test Is Equivalent to the Wilcoxon Rank-Sum

question 3

Essay

The Mann-Whitney U test is equivalent to the Wilcoxon rank-sum test for independent samples in the sense that they both apply to the same situations and always lead to the same conclusions. In the Mann-Whitney U test we calculate $$z = \frac { U - \frac { \mathrm { n } _ { 1 } \mathrm { n } _ { 2 } } { 2 } } { \sqrt { \frac { \mathrm { n } _ { 1 } \mathrm { n } _ { 2 } \left( \mathrm { n } _ { 1 } + \mathrm { n } _ { 2 } + 1 \right) } { 12 } } }$$
where
$$\mathrm { U } = \mathrm { n } _ { 1 } \mathrm { n } _ { 2 } + \frac { \mathrm { n } _ { 1 } \left( \mathrm { n } _ { 1 } + 1 \right) } { 2 } - \mathrm { R }$$ For the sample data below, use the Mann-Whitney U test to test the null hypothesis that the two independent samples come from populations with the same distribution. State the hypotheses, the value of the test statistic, the critical values, and your conclusion.
Test scores (men): 70, 96, 77, 90, 81, 45, 55, 68, 74, 99, 88
Test scores (women): 89, 92, 60, 78, 84, 96, 51, 67, 85, 94

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