Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

APPENDIX E | Hypothesis Tests for Ordinal Data: Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman Tests 691
Notice that these are the same U values we obtained in Example E.1 using the counting
method. The Mann-Whitney U value is the smaller of these two, U 5 6.
■ Evaluating the Significance of the Mann-Whitney U
Table B9 in Appendix B lists critical value of U for a 5 .05 and a 5 .01. The null
hypothesis is rejected when the sample data produce a U that is less than or equal to the
table value.
In Example E.1 both samples have n 5 6 scores, and the table shows a critical value
of U 5 5 for a two-tailed test with a 5 .05. This means that a value of U 5 5 or smaller
is very unlikely to occur (probability less than .05) if the null hypothesis is true. The data
actually produced U 5 6, which is not in the critical region. Therefore, we fail to reject
H 0
because the data do not provide enough evidence to conclude that there is a significant
difference between the two treatments.
There are no strict rules for reporting the outcome of a Mann-Whitney U-test. However,
APA guidelines suggest that the report include a summary of the data (including
such information as the sample size and the sum of the ranks) and the obtained statistic
and p value. For the study presented in Example E.1, the results could be reported as
follows:
The original scores were ranked ordered and a Mann-Whitney U-test was used to compare the
ranks for the n 5 6 participants in treatment A and the n 5 6 participants in treatment B. The
results indicate no significant difference between treatments, U 5 6, p . .05, with the sum of
the ranks equal to 27 for treatment A and 51 for treatment B.
■ Normal Approximation for the Mann-Whitney U
When the two samples are both large (about n 5 20) and the null hypothesis is true,
the distribution of the Mann-Whitney U statistic tends to approximate a normal shape.
In this case, the Mann-Whitney hypotheses can be evaluated using a z-score statistic
and the unit normal distribution. The procedure for this normal approximation is as
follows.
1. Find the U values for sample A and sample B as before. The Mann-Whitney
U is the smaller of these two values.
2. When both samples are relatively large (around n 5 20 or more), the distribution of
the Mann-Whitney U statistic tends to form a normal distribution with
m5 n A n B
2
and
s5Î n A n B sn A 1 n B 1 1d
12
The Mann-Whitney U obtained from the sample data can be located in this distribution
using a z-score:
z 5 X 2m
s
5
U 2 n n A B
Î
2
n n sn 1 n
A B A B
1 1d
12
3. Use the unit normal table to establish the critical region for this z-score. For
example, with a 5 .05, the critical values would be ±1.96.
Usually the normal approximation is used with samples of n 5 20 or larger; however,
we will demonstrate the formulas with the data that were used in Example E.1. This study