ideal tests for experimental economics (man whitney etc)

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ideal tests for experimental economics (man whitney etc)

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Dear Statalisters,
I have done an economic experiment and  want to compare the two groups in terms of median, mean and whether they are coming from the same population etc.
I definitely do not feel comfortable about my knowledge about econometrics, so i need help about first what kind of tests are appropriate for my issue and how to interprete these tests.
I really always confused about the wording in econometrics, such as ' we fail to reject the null hypothesis'
So why dont we say that the null hypothesis is true, etc.
In my study:
Group A sample size is 17
Group B sample size is 18.
What can you suggest me about possible tests? which one to choose?
I did two of them allready,do you have more suggestions sich as KS test etc.
And i dont know how to interprete the results, I looked at the manuels but really i cant be sure whether I understood it correctly or not.
first one is median test, what is the formal name of this test in the literature, I can see from the command that it is testing the median but is there a formal name of the test which is related to the founders of the test, such as Man whitney.
1.). median offer, by(buyer)

Median test

   Greater |
  than the |         buyer
    median |         0          1 |     Total
        no |         9         10 |        19
       yes |         9          7 |        16
     Total |        18         17 |        35

          Pearson chi2(1) =   0.2743   Pr = 0.600

   Continuity corrected:
          Pearson chi2(1) =   0.0340   Pr = 0.854

What does the Pearson and continutity corrected tell me, are the two medians different or not.
Which one should I look, continutity correction or the one above.

2.)Another one is ranksum, but is this test same with U test, I am confused about the names, and the result interpretation

 ranksum offer, by(buyer)

Two-sample Wilcoxon rank-sum (Mann-Whitney) test

       buyer |      obs    rank sum    expected
           0 |       18         329         324
           1 |       17         301         306
    combined |       35         630         630

unadjusted variance      918.00
adjustment for ties       -9.26
adjusted variance        908.74

Ho: offer(buyer==0) = offer(buyer==1)
             z =   0.166
    Prob > |z| =   0.8683

What does the results say, are they coming from different distributions or not?
the means are different or not?

Thank you very much in advance.