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t-test of a correlation coefficient is used for testing the sample correlation against population correlation of zero. To test an assumed value of the population coefficient other than zero, we should use z-test for a correlation coefficient.

Question:

  1. What makes the tests different such that the t-test can only test for zero? Isn't the z-test a special case of the t-test (infinite degree of freedom)?

  2. Since the t-test can't be used for tested for non-zero correlation. Why can't we always use the z-test? What's the point of the t-test?

EDITED for reference:

100 Statistical Tests by Gopal K.Janji

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Short answer: You don't use a t-test of the depicted form when $\rho\neq 0$ simply because the statistic $\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$ has a $t_{n-2}$ distribution when $\rho= 0$ (and the other assumptions hold) and it doesn't have a $t_{n-2}$ distribution otherwise.

There's several different z-tests for a correlation coefficient (whether $\rho$ is $0$ or not), but they're approximate, large sample tests. Some approximations are better than others. It may be that you can also make a good t approximation, but I don't recall seeing a derivation of any that would give you a suitable $\nu$.

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