Which test for two groups of samples (unpaired, different size)?

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I need a little help with this problem. I work in medicine and have two different groups of patients.

two groups of samples (unpaired and different size):

a= 67,4,37,91,2,3,87,2,9,3,67,38,195,0,94,1,0,3,2,3

b= 3,188,91,10,89,3,3,21,5,3,73,123,73,2,33,3,60,5,69,22,0,31,5,1,4,1,26,0,3,139,66,27,103,51,120,4,10,27,45,153,1,95

Many studies have shown, that there is significant difference between these groups of patients, or at least it should be.....but not in my tests. I tried standard t-test, but it says there is no significant difference between groups (p=0,60), which is pretty bad for me.

I need a test which shows, that those two groups have significant difference, because I don´t know if the t-test or Welch´s test was good choice for this.

In matlab:

[h,p,ci,stats] = ttest2(a,b)

h = 0

p = 0.5960

ci = -34.4264 19.9407

stats = struct with fields: tstat: -0.5330 df: 60 sd: 50.0214

Is the t-test (Welch's t-test) I used a good choice? I assume the groups should have the same mean (they haven't).

Which test should I use in matlab to answer the question about significant difference? Any ideas?

Thank you a lot :)