Could anyone confirm some dubious assumptions about the t-test?

Question

For the most part, all the sources I have come across blankly state that the unpaired t-test assumes equal variances among the two groups, and that the data has a normal distribution.

Now, some sources go further and state that the t-test should not be used for small sample sizes ( < 8), but that parametric tests should be used instead (specifically Mann Whitney U test). The sources continue to say that the sensitivity to the normality assumption is valid up to 40 samples, where after it's not really that sensitive. I haven't been able to confirm this (the 40 samples) from any other sources. Could anyone confirm this statement?

I know the central limit theorem (CLT) maintains that samples larger than 30 start approximating the normal curve. However, performing the Anderson-Darling test on over 100 samples still confirm that the data is non-normal. Will performing an unpaired t-test on these "guaranteed" non-normal data still be valid? Do people just blindly apply the CLT assumption?

Answer 1

I think you can answer your own question. Get a reliable stat text with the Z and t tables. Find the number of samples in the t table that are "close enough" to comparable values in the Z table. Use that sample size as the size you'd need to treat the sample as approximately Normal. (You can also do this mathematically...look up the design of experiments in a good text.)

I've seen 20-40 as sample sizes in a variety of text (I have dozens). But for me, 30 is what I use...all else equal. Also, n < 8 is clearly not a good sample size for most things. But the t test takes that into account with the corresponding wider intervals of deviation. More samples will certainly give you narrower intervals of spread.

By the way, have you looked at the 2-way ANOVA for your small samples if you are uncertain about the variances of each treatment/group?

I don't blindly follow the CLT...or any other assumed frequency distribution. If there are reasons for assuming something else, I will. But the CLT does seem to hold in the overwhelming majority of cases I've handled as a systems analyst and nuclear research associate. The more...the merrier (and more normal).