Question

Using the Mann-Whitney U Test 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

\(z = \frac{{U - \frac{{{n_1}{n_2}}}{2}}}{{\sqrt {\frac{{{n_1}{n_2}\left( {{n_1} + {n_2} + 1} \right)}}{{12}}} }}\)

Where

\(U = {n_1}{n_2} + \frac{{{n_1}\left( {{n_1} + 1} \right)}}{2} - R\)

and R is the sum of the ranks for Sample 1. Use the student course evaluation ratings in Table 13-5 on page 621 to find the z test statistic for the Mann-Whitney U test. Compare this value to the z test statistic found using the Wilcoxon rank-sum test.

Short answer

The test statistic value of the Mann-Whitney U test is equal to 0.41.

The test statistic value of the Wilcoxon rank-sum test is equal to -0.41.

The values of z for both the tests are the same but with opposite signs.

Step-by-step solution

Given information

The Wilcoxon rank-sum test and the Mann-Whitney U test arenon-parametric tests used to test the difference of medians between two independent samples when both tests are applied to similar situations.

The data are given on course evaluation ratings for courses taught by male and female professors.

Assign ranks and then calculate the test statistics for the Mann-Whitney U test

The ranks of the observations from the two samples are given using the following steps:

• Combine the two samples and label each observation with the sample name/number it comes from.
• The smallest observation is assigned rank 1; the next smallest observation is assigned rank 2, and so on until the largest value.
• If two observations have the same value, the mean of the ranks is assigned to them.

The following table shows the ranks:

\(\)

Evaluation ratings	Sample name	Ranks
4.3	Female	20.5
4.3	Female	20.5
4.4	Female	23.5
4.0	Female	13.5
3.4	Female	5.5
4.7	Female	27
2.9	Female	1
4.0	Female	13.5
4.3	Female	20.5
3.4	Female	5.5
3.4	Female	5.5
3.3	Female	3
4.5	Male	25.5
3.7	Male	8
4.2	Male	17.5
3.9	Male	11
3.1	Male	2
4.0	Male	13.5
3.8	Male	9.5
3.4	Male	5.5
4.5	Male	25.5
3.8	Male	9.5
4.3	Male	20.5
4.4	Male	23.5
4.1	Male	16
4.2	Male	17.5
4.0	Male	13.5

The sum of the ranks corresponding to female professors is equal to:

\(\begin{array}{c}R = 20.5 + 20.5 + 23.5 + .... + 3\\ = 159.5\end{array}\)

Let\({n_1}\)be the sample size corresponding to the female professors.

Let\({n_2}\)be the sample size corresponding to the male professors.

Here,

\(\begin{array}{l}{n_1} = 12\\{n_2} = 15\end{array}\)

The value of U is computed as follows:

\(\begin{array}{c}U = {n_1}{n_2} + \frac{{{n_1}\left( {{n_1} + 1} \right)}}{2} - R\\ = \left( {12 \times 15} \right) + \frac{{12\left( {12 + 1} \right)}}{2} - 159.5\\ = 98.5\\\end{array}\)

The value of the z statistic for the Mann Whitney U test is as follows:

\(\begin{array}{c}z = \frac{{U - \frac{{{n_1}{n_2}}}{2}}}{{\sqrt {\frac{{{n_1}{n_2}\left( {{n_1} + {n_2} + 1} \right)}}{{12}}} }}\\ = \frac{{98.5 - \frac{{12 \times 15}}{2}}}{{\sqrt {\frac{{\left( {12 \times 15} \right)\left( {12 + 15 + 1} \right)}}{{12}}} }}\\ = 0.41\end{array}\)

Thus, z is equal to 0.41.

Calculate the mean, standard deviation, and test statistic for the Wilcoxon rank-sum test

The mean value\(\left( {{\mu _R}} \right)\)is as follows:

\(\begin{array}{c}{\mu _R} = \frac{{{n_1}\left( {{n_1} + {n_2} + 1} \right)}}{2}\\ = \frac{{12\left( {12 + 15 + 1} \right)}}{2}\\ = 168\end{array}\)

The standard deviation\(\left( {{\sigma _R}} \right)\)is as follows:

\(\begin{array}{c}{\sigma _R} = \sqrt {\frac{{{n_1}{n_2}\left( {{n_1} + {n_2} + 1} \right)}}{{12}}} \\ = \sqrt {\frac{{12 \times 15\left( {12 + 15 + 1} \right)}}{{12}}} \\ = 20.49\end{array}\)

The test statistic value for the Wilcoxon rank-sum test is calculated as follows:

\(\begin{array}{c}z = \frac{{R - {\mu _R}}}{{{\sigma _R}}}\; \sim N\left( {0,1} \right)\\ = \frac{{159.5 - 168}}{{20.49}}\\ = - 0.41\end{array}\)

Thus, z is equal to -0.41.

Comparison

The values of the test statistic (z) arethe same for both the tests, i.e.,equal to 0.41,but the signs are opposite.

The value of z for the Mann Whitney U test is positive, whereas the value of z for the Wilcoxon rank-sum test is negative.