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


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
MannWhitney U Test, which is the nonparametric equivalent of
the \(t\)
test for means.
The \(U\)test (as the majority of nonparametric tests) uses the rank sums of the two samples. 

Procedure 
The test is implemented as follows.


Null Hypothesis  The null hypothesis is: the two populations have the same central tendency. The alternative hypothesis is: The central tendencies are NOT the same.  
Test statistic 
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:
$$ z = \frac{U  \mbox{E }(U)}{\sigma} \, , $$
where
$$\mbox{E }(U) = 0.5 \, n_1 \, n_2 \,\,\,\,\, \mbox{ and } \,\,\,\,\,
\sigma^2 = \frac{n_1 \, n_2 \,(n_1 + n_2 + 1)}{12} \, . $$
The critical value is the normal tabled \(z\)
for \(\alpha/2\)
for a twotailed test or \(z\)
at \(\alpha\)
level, for a onetail test.
For small samples, tables are readily available in most textbooks on nonparametric statistics. 

Example  
An illustrative example of the \(U\) test 
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 central tendencies 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 the large sample approximation.
For \(U = 40.0\) and \(\mbox{E }(U) = 0.5 \, n_1 \, n_2 = 60.5\), the test statistic is $$ z = \frac{U  \mbox{E }(U)}{\sigma} = \frac{40.0  60.5}{15.23} = 1.346 \, , $$ where $$ \sigma = \sqrt{\frac{n_1 \, n_2 \, (n_1 + n_2 + 1)}{12}} = \sqrt{\frac{11(11)(11+11+1)}{12}} =15.23 \, . $$ For a twosided test with significance level \(\alpha\) = 0.05, the critical value is \(z_{1\alpha/2}\) = 1.96. Since \(z\) is less than the critical value, we do not reject the null hypothesis and conclude that there is not enough evidence to claim that two groups have different central tendencies. 