From what size considering that the sample is large and allows to use the parametric tests

Question

With a large sample size we can used a parametric test whithout testing a normality of distribution. from what size can we consider that the size of the sample is large?

Answer 1

1. Not all parametric tests assume normality. Some make other parametric distributional assumptions.

   A test might assume uniform distributions or exponential distributions or any number of other possibilities. Parametric but not normal.

2. There can be no single sample size that suffices to achieve reasonable approximation of significance level for all parametric tests and all distributions.

   Even if we restrict consideration to only tests that assume normality, no matter how large a sample size you choose there's always cases for which that will not be sufficient.

   Consider the F-test for equality of variances

   It's a parametric test that assumes normality.

   Now consider sampling from a pair of exponential distributions with equal sample sizes. It doesn't matter how large that sample size gets, that F-test won't have an F-distribution, or even anything close to it.

   Even if we just restrict ourselves to t-tests, there's no single sample size that will always be sufficient. Let's say we look at a one sample t-test and you pick any sample size you like; I'll tell you a distribution for which that $n$ isn't large enough.

3. Even if the conditions are such that the test has about the right significance level, the power may be quite poor compared to obvious alternative choices of test. Large samples can't help with poor asymptotic relative efficiency; if you have a large sample size because you're trying to pick up a small effect, you may need a test with better power in that circumstances.

But in any case I would not usually advise testing normality in order to use a hypothesis test that assumes it.