When the distribution is known, should I use t-test or z-test for testing the significance of Pearson Correlation Coefficient?

Question

Generally speaking, t-test is used for testing the pearson correlation coefficient significance of X's and Y's. In my case, however, assuming X and Y both follow some known distributions. I have a series of data (x1, y1), (x2, y2),..., (xn, yn) drawn from X and Y. If I want to test the significance of the pearson correlation coefficient of X's and Y's, should I use z-test since distribution is known or use t-test since it is the default?

Answer 1

Testing the significance of a correlation is akin to performing a regression of y on x and evaluating the coefficient of x in the model. The reason you use a t-test in that scenario is because we don't know the variance of $y\vert x$; It must be estimated.

I would suppose that if you knew the variance of $y \vert x$ a priori , you could use a z test.

But honestly, if $n$ is large enough, there will not be a huge difference between the two.