I have done so many research and have read so many posts and manuscripts to find an answer to my question but I'm getting more and more confused. So I found it best to ask my question directly.
As we all know, many statistical test have the assumption of normality. let's think about ANOVA. there doesn't normality mean that in each group the observations must be normally distributed, but the sampling distribution of the means must be normal.
To clarify this, if you take a sample of size k and calculate its mean, then repeat this procedure by sampling n times each with size k, then the means of all these n samples of size k should be normally distributed. and the sample size k must be at least 30 for this to be held in worst scenarios. That is the central limit theorem.
1- So when performing ANOVA, if the sample size in each group is above 30, we know that the central limit theorem is held. That is we don't need to check for assumption of normality. right?
2- But if the sample size in each group is less than 30, If we know the population where the sample comes from is normal (Like we know that blood pressure is always normal), then we don't need to check the normality assumption. Even if Shapiro wilk or other tests claim that the sample is not normal, we still can assume the data as normal since the population is widely known as normal.
3- But if the sample size is below 30 and we don't know what distribution of the population where the data comes from, we conduct the Shapiro Wilk or other relevant tests. right?
4- And if the data is count and obviously coming from Poisson or NB distribution, or if we already know the population is not normal, there is no need to run shapiro wilk and we go for non-parametric tests?
***** Does anybody know if one variable is count and and the other variable is normal, Spearman corr test is suitable?