One of the assumptions for t-tests is that the data must follow a normal distribution.
However, due to the Central Limit Theorem (and this thread): "if the sample is large enough you can use t-test (with unequal variances)". I'm trying to sort out what this means for my case. I think my sample should be large-enough, but how to confirm it?
A Levene's test showed that the two samples don't have an equal variance, hence I plan to use Welch's test (the unequal variance version of the t-test). I've also ran the Shapiro-Wilk test to confirm that one of my two samples doesn't, in fact, follow a normal distribution.
I need to run the tests for a few different cases, but to keep things short I'm detailing only two of them.
Sample sizes are 19 and 15, respectively for group1 and group2 (this happens on both the examples: Case1 and Case2).
Results of Shapiro-Wilk's test for normality
Case1 sample | p_value | w | Result group1 | 0.104 | 0.918 | Normal group2 | 0.027 | 0.863 | Not Normal (p<0.05) Case2 sample | p_value | w | Result group1 | 2.054e-05 | 0.663 | Not Normal (p<0.05) group2 | 0.006 | 0.814 | Not Normal (p<0.05)
Results of Levene's test for equality of variances
Case1 p_value | w | Result 0.154 |2.128 | Equal Variance Case2 p_value | w | Result 0.0251 |5.521 | Unequal Variance (p<0.05)
Result of the one-tailed (Welch) t-test (H1: group1>group2)
Case1 t_statistic | p_value | Result 3.073 | 0.002 | Significant (p<0.05) Case2 t_statistic | p_value | Result 2.475 | 0.012 | Significant (p<0.05)