Mann-Whitney U Test

1. mixing and sorting .
2. ranking;
   1. each element receives a rank corresponding to its position in the list.Element whihc have the same value receive the same rank:
   2. 
   3. 
3. compute rank sums R_a,R_b
   1. like this.
   2. for these sums,the following always holds:
4. compute sample statistics U_a,U_b
   1. like this.
   2. where the following always holds: U_a+U_b=n_an_{b------>7+41=6*8=48}
5. set U=min {U_a,U_b}
   1. U=min {U_a,U_b}=min{7,41}=7
6. compute critical U_α values
   1. for the significant level α the critical U_α value s can be computed for the two-sided test as:
   2. where z is the "probity "function ,the inverse cumulative distribution function of the standard normal distribution .
   3. the value of z can be looked up in the standard normal distribution table appendix .
7. U < U_α => different between U_a and U_b significant 
   1. the different between U_a and U_b is significant at an error level a only if U is smaller than U_a.
   2. if U < U_a,and U_a < U_b:A is from a distribution with a smaller median than B (this is wrong with a probability of no more than a )
   3. 
   4. for this example :