I need to test to compare the means of two samples, one with size 21 and the other with size 47. Histograms show Sample 1 is skewed to the right while Sample 2 has a bell symmetrical shape. Descriptive statistics will be found below. I tested for normality with Shapiro-Wilk any way, rejecting the null hypothesis for either, so that will confirm they are not normal. I understand t-test will not be appropriate to this case. I moved to non-parametric and used Wilcoxon rank sum test and got p-value 0.2395 which will indicate there is no evidence to think the means are different (they are equal).
Is it correct to use Wilcoxon rank sum in this case where the shape of both samples is different? Am I missing something? Or with this information can I state that there is no difference between the means? Is it necessary to do something else like permutations?
Sample 1 Shapiro-Wilk p-value = 0.000 average = 0.0270 sd = 0.0892 kurtosis 10.4 skewness 3.2 n = 21 Sample 2 Shapiro-Wilk p-value = 0.0366 average = 0.0367 sd = 0.0752 kurtosis 0 skewness 0.6 n = 46