7.3.5. Do two arbitrary processes have the same mean?

7. Product and Process Comparisons
7.3. Comparisons based on data from two processes

7.3.5. Do two arbitrary processes have the same mean?

The nonparametric equivalent of the t test is due to Mann and Whitney, called the U test

By "arbitrary" we mean that we make no underlying assumptions about normality or any other distribution. The test is called the Mann-Whitney U-Test, which is the nonparametric equivalent of the t-test based for normal means.

The U-test (as the majority of nonparametric tests) uses the rank sums of the two samples. The procedure flows as follows

Procedure

Rank all (n₁ + n₂) observations in ascending order. Ties receive the average of their observations.
Calculate the sum of the ranks, call these T_a and T_b
Calculate the U statistic, U_a = n₁(n₂) + .5(n₁)(n₁ + 1) - T_a
or
U_b = n₁(n₂) + .5(n₁)(n₁ + 1) - T_b where U_a + U_b = n₁(n₂).

The null hypothesis is: the populations have the same median. The alternative hypothesis is: The medians are NOT the same.

The test-statistic, U, is the smaller of U_a and U_b. For sample sizes larger than 20, we can use the normal z as follows:

z = [ U - E(U)] / s where

The critical value is the normal tabled z for a/2 for a two tailed test or for a z at a level, for a one tail test.

For small samples use tables, which are readily available in most textbooks on nonparametric statistics.

An illustrative example of the U test

Example

Two processing systems were used to clean wafers. The following data represent the (coded) particle counts. The null hypothesis is that there is no difference between the means of the particle counts; the alternative hypothesis is that there is a difference. The solution shows the typical kind of output software for this procedure would generate, based on a the large sample approximation approach.

Group A Rank Group B Rank

.55 8 .49 5

.67 15.5 .68 17

.43 1 .59 9.5

.51 6 .72 19

.48 3.5 .67 15.5

.60 11 .75 20.5

.71 18 .65 13.5

.53 7 .77 22

.44 2 .62 12

.65 13.5 .48 3.5

.75 20.5 .59 9.5

N Sum of Ranks U Std. Dev of U Median

A 11 106.000 81.000 15.229 0.540

B 11 147.000 40.000 15.229 0.635

Enter value for a (press Enter for .05): .05
Enter 1 or 2 for One or Two sided test: 2

E(U) = 60.500000

The Z-test statistic = 1.346133
The critical value = 1.960395.

Probability of Z-test = 0.910870
Right Tail Area = 0.089130

Cannot reject the null hypothesis.

A two-sided confidence about U - E(U) is:

Prob {-9.3545 < DELTA < 50.3545 } = 0.9500

DELTA is the absolute difference between U and E(U).
The test statistic is given by: (DELTA / SIGMA).