Assume I have a pair of vectors $a,b$. Upon discussing their correlation $\rho_{a,b}$ we can usually test whether $\rho_{a,b}=0$, $\rho_{a,b}>0$ or $\rho_{a,b}<0$ (as asked previously here and here). These tests can be easily conducted in R using cor.test. For Pearson's $\rho$ we even have a specified distribution of the test statistic under the null hypothesis on independence:
$$t=\rho\sqrt{\frac{n-2}{1-\rho^2}}\sim t_{n-2}$$
Now, assume I would like to test for a strong positive correlation. That is, my null hypothesis would be $H_0:\rho_{a,b}\ge 0.7$. What would be the distribution of the test statistic? Is there a way to conduct this using cor.test?
