I am hoping to get some clarification on how to interpret what is often listed in statistical textbooks as an assumption of the independent samples t-test: Data are normally distributed.
I had assumed that this normality assumption was in reference to the shape of the distribution of the variable of interest (the dependent Variable) in the population -- not the shape of the sampling distribution. However, based on some recent comments that I have read, it sounds as this often cited assumption is in fact about the shape of the sampling distribution and NOT the shape of the variable of interest. Is this the case? If so, and we know that our sample size in both groups is greater than 30 subjects per group, why bother to assess normality given that one can invoke the central limit theorem?
Alternatively, is the normality assumption of the t-test about the shape of the actual distribution in the population but the idea is that the actual p-values associated with the t-test are valid because of CLT and the idea that the shape of the sampling distribution is normal with sample over 30, regardless of the shape of the sample distribution.