I understand that, in the case of a highly skewed population and sample, the sampling distribution of the mean can still be normally distributed if the sample size is large, according to Central Limit Theorem. This means that, arguably, the normality assumption of a t-test holds despite a skewed sample, because t-tests require that the sampling distribution is normal.
However, we also know that the Mean statistic (x bar) can be affected by extreme scores, hence the use of the median in national average salary data.
If the mean is misleading, can it still be used in t-tests, even if CLT suggests the normality assumption (of the sampling distribution of the sample mean) is met? Or is it the case that the standard error of the mean is robust to sample skewness, according to CLT, and thus we can be confident that our confidence intervals are accurate? Could a type II error not occur, whereby two populations appear equal because one has a highly skewed population that biases the mean, even if the two populations have different underlying central tendencies?