I haven't been able to find a definitive answer to my question in previous topics, so I'll post it here. I am sure this is a trivial question for many, but I do not have ny formal training in statistics/mathematics, and I would like to have a deeper understanding when I apply statistical methods to my research data.
So, from what I understand, the sampling distribution of the t-statistic is the t-distribution defined by the appropriate degrees of freedom. I also understand that the sampling distribution of the difference between two means follows a normal distribution if the sample sizes are big enough, and that we use the t-distribution to approximate the population distribution when the population variance is unknown or when the sample size is too small to reasonably assume that the Central Limit Theorem will kick in. However, I do not understand why we need to concern ourselves with the assumption of normality if what we are concerned with is the sampling distribution of the test statistic, and we know the test statistic to follow a t-distribution with the appropriate degrees of freedom. I think I'm missing a crucial link somewhere, and I would be glad if someone could offer a non-technical explanation of what is it that I am misunderstanding.
Thank you for the feedback!