In the textbooks I have access to (and that discuss hypothesis testing for correlation), I only met examples, where the null-hypothesis was $\rho=0$, and the alternative hypothesis was $\rho\ne 0$. My question is about using a one-sided alternative hypothesis $\rho>0$. Is this meaningful?
This question has been asked before, but it has not been answered. There was a comment next to the linked question, that said that the null-hypothesis should be $\rho\le 0$ in case we would like a one-sided alternative hypothesis, but I have problems with this comment. As I understand, the t-distribution that is used for testing the correlation coefficient is only valid when $\rho=0$, so we have no choice, but using this as the null-hypothesis.
So, to summarize: can we test $H_0:\rho=0$ against $H_1:\rho>0$ using $R\sqrt{\dfrac{n-2}{1-R^2}}$ and the t-distribution with degree of freedom $n-2$?
Best Answer
Yes. Instead of using a two-sided critical value from a t-distribution with $n-2$ degrees of freedom (e.g., $\pm 2.09$ for $n=22$ and $\alpha=.05$, two-sided), you would use just the upper critical value (e.g., $+1.72$ for $n=22$ and $\alpha=.05$, one-sided).