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My data has 150 samples and is not normally distributed, and it does not have equal variances either. I know t-test (e.g. this python method) can be used when data is normally distributed and it can be used when it does not have equal variance. So I am not sure if I can use the t-test when data is not normally distributed.

Alternatively, I thought, I should use a non-parametric test, e.g., the Mann-Whitney $U$ rank sum test, but as far as I know, the Mann-Whitney $U$ can be used only if the samples have equal variance!

My question is, am I correct, can't I use any of these test (t-test and Mann-Whitney $U$)? If not, which test should I use?

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The Mann Whitney U test does not assume that there are equal variances. However, it tests different hypotheses depending on whether the two distributions have the same shape or not. In your case, where the shapes are different, it tests whether the distributions are different (see Laerd Statistics, although one might ask how they can be the same if they have different shapes (this gets into statistical significance).

If you want to test whether your two distributions have the same mean and not make any assumptions, you could do a permutation test.

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    $\begingroup$ Same shape and same variance. $\endgroup$ – Alexis Feb 23 at 21:23
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I run into a similar problem recently. I needed to quantify the presence of statistical significance, but I could neither apply the t-test nor the Wilcoxon signed rank test.

In the end, I went for confidence intervals constructed by the bootstrap. It is easy to apply and makes very few assumptions about the underlying distribution.

The Wikipedia article is a great place to start (link).

A good book on this is the following: A. C. Davison, D. V. Hinkley et al., Bootstrap methods and their application. Cambridge university press, 1997, vol. 1

Also, in the off chance you also use Julia, there is a great package implementing all this (link).

Hope you find this useful.

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