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I a have two independent samples one with more than 700 observations and the second with more than 500 observations. I firstly use Cramér-von Mises test to test normality just to check how non-normal the distribution of the datasets is. It returned me p-value equal to 0. Thus, I have to rely on CLT (hope that I have enough observations - or should I rather somehow "test" that?). I then used Conover test to test equality of variances and it again returned me p-value equal to almost zero. Therefore, I want to compute the unequal variance t-test (Welch test), but I am wondering whether it was better to used standard Student's t-test because as I understand that due to the Lindenberg CLT, we can disregard the unequality of variances. So, generally, which test should I use? And was the above-mentioned testing of data reasonable?