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The Mann-Whitney U-test is used to test whether two independent samples of observations are drawn from the same or identical distributions. An advantage with this test is that the two samples under consideration do not necessarily need to have the same number of observations or instances.
Yes, this sounds like a repeated measures ANOVA with one repeated measures Time factor (before/after) and one Treatment factor. You can test for normality using Shapiro-Wilk and for homogeneity of variances using Leveneâs test. You can use an epsilon correction instead of testing for sphericity.