from what I understand the assumption of normality (that must be assumed if one wants to use parametric tests) refers to the sampling distribution of the mean. It does not imply that the distriubtion is normal in the population itself nor a in a given sample. With the CLT we can assume that with a large enough sample this assumption is met:
given random and independent samples of N observations each, the distribution of sample means approaches normality as the size of N increases, regardless of the shape of the population distribution
Therefore (from my understanding):
If we have a large sample ( I know the N > 30 is debatable, but say the N = 500 simple random sample) OR the variable of interest form population that we are sampling from is known to be normal, then the sampling distribution has to be normal and therefore we can use parametric tests. For example I stumbled across this in one of the scientific journals:
With large enough sample sizes (> 30 or 40), the violation of the normality assumption should not cause major problems (4); this implies that we can use parametric procedures even when the data are not normally distributed (8). If we have samples consisting of hundreds of observations, we can ignore the distribution of the data (3). According to the central limit theorem, (a) if the sample data are approximately normal then the sampling distribution too will be normal; (b) in large samples (> 30 or 40), the sampling distribution tends to be normal, regardless of the shape of the data (2, 8); and (c) means of random samples from any distribution will themselves have normal distribution (3). Link to the article
Going back to my question, if the normality assumption DOES NOT imply that the variable of interest in a sample is normaly distributed (only that the sampling distriubtion of the mean is normal). why do people plot histograms to see if the data in a sample is normally distributed? How is this relevant for the assumption of normality?