I have calculated the correlation $\rho$ between $X$ and $Y$. I'd like to test $H_0: \rho \leq r$, $H_1: \rho > r$.

For the case when $r = 0$, I can apply the transform $t = \rho \frac{\sqrt{n-2}}{\sqrt{1 - \rho^2}}$, which follows the $t_{n-2}$ distribution. I can then perform a one-tailed test and see if $\int_t^{\infty} p_{t_{n-2}} < \alpha$, rejecting the null hypothesis if it is.

How can I generalize this case to the described scenario when $r \not= 0$?

Edit: how would I determine the sample size required to perform this test with specified $\alpha$ and $\beta$ (1 - power)?

I have calculated the correlation $r$ between $X$ and $Y$. I'd like to test $H_0: r \leq \rho$, $H_1: \rho < r$.

For the case when $r = 0$, I can apply the transform $t = \rho \frac{\sqrt{n-2}}{\sqrt{1 - \rho^2}}$, which follows the $t_{n-2}$ distribution. I can then perform a one-tailed test and see if $\int_t^{\infty} p_{t_{n-2}} < \alpha$, rejecting the null hypothesis if it is.

How can I generalize this case to the described scenario when $r \not= 0$?

Edit: how would I determine the sample size required to perform this test with specified $\alpha$ and $\beta$ (1 - power)?

I have calculated the correlation $\rho$ between $X$ and $Y$. I'd like to test $H_0: \rho \leq r$, $H_1: \rho > r$.

For the case when $r = 0$, I can apply the Fisher transform $t = \rho \frac{\sqrt{n-2}}{\sqrt{1 - \rho^2}}$, which follows the $t_{n-2}$ distribution. I can then perform a one-tailed test and see if $\int_t^{\infty} p_{t_{n-2}} < \alpha$, rejecting the null hypothesis if it is.

How can I generalize this case to the described scenario when $r \not= 0$?

Edit: how would I determine the sample size required to perform this test with specified $\alpha$ and $\beta$ (1 - power)?

I have calculated the correlation $\rho$ between $X$ and $Y$. I'd like to test $H_0: \rho \leq r$, $H_1: \rho > r$.

For the case when $r = 0$, I can apply the transform $t = \rho \frac{\sqrt{n-2}}{\sqrt{1 - \rho^2}}$, which follows the $t_{n-2}$ distribution. I can then perform a one-tailed test and see if $\int_t^{\infty} p_{t_{n-2}} < \alpha$, rejecting the null hypothesis if it is.

How can I generalize this case to the described scenario when $r \not= 0$?

Edit: how would I determine the sample size required to perform this test with specified $\alpha$ and $\beta$ (1 - power)?