# Mann Whitney U test with normal distribution approximation: null hypothesis rejected?

I'm new with U test and I have some doubts about the rejection of the null hypothesis with the U test with normal distribution approximation.

In my example I used this data for a 1 tailed test:

$$H_0: median_1 = median_2$$ $$H_1: median_1 < median_2$$ $$\alpha = 0.05$$ $$Z_\alpha = 1.645$$

I obtained $$Z = 1,0313$$

Do I reject the null hypothesis if $$Z < Z_\alpha$$ or if $$Z > Z_\alpha$$?

I'm new with the U test and I have some doubts about the rejection of the null hypothesis with the U test with normal distribution approximation.

H0:median1=median2
H1:median1$<$median2

The Mann-Whitney U test is not of itself a test of the hypothesis of equality of population medians -- at least no more than it is a test of equality of population means. If accompanied by additional assumptions (that would make what it does test equivalent to a test of equality of populations) then it can function as such a test.

I obtained Z=1,0313

Do I reject the null hypothesis if $Z<Z_α$ or if $Z>Z_α$

Possibly neither of those -- it depends on exactly what the statistic you have is computing. Note, however, that it would be very unusual indeed to have a test where the rejection region included a Z-score of 0 and this is not one of those rare cases.

If your $U$ is the number of times an observation from sample 1 exceeds an observation from sample 2 (what I'd expect to be the most likely definition of $U$) then you'd reject for unusually small values of the statistic, so most likely you'd actually reject for $Z\leq -Z_α$.

However, you should double check how $U$ has actually been defined for your situation (and strictly speaking, you should also double check how $Z$ is defined in terms of that $U$ as well). That would then confirm which direction the comparison should go (i.e. whether the rejection region is $Z\leq -Z_α$ or $Z\geq Z_α$).

U test can be explained with Area Under ROC see this post. It is a non-parametric test, not necessary to tell if mean or median is different but two distributions are different.

Do not know how if you like to try following code to see how the numbers are calculated.