I'm currently studying 'Estimation of population correlation coefficient from sample correlation coefficient. Source: https://newonlinecourses.science.psu.edu/stat501/node/259/
Test statistic is like below.
$t = r*\sqrt(N-2)/(\sqrt(1-r^2)$
N is the size of sample and r is correlation coefficient of the sample. The statistic t is known to follow Student t-distribution.
According to the source above, it adopts two-tailed test with below null/alternate hypothesis.
H0: population correlation coefficient $\rho$ = 0
H1: population correlation coefficient $\rho$ != 0; $\rho$ > 0 or $\rho$ < 0.
To me, H1 above makes sense because I can't find any directional/sign information from the test statistic.
But today, my colleague said to me that I should adopt one tailed test. This is because we have a background knowledge that the population correlation coefficient should be always smaller than 0.
So, here're my questions.
Does the test statistic above give me any information about sign of $\rho$?
In case I have a background knowledge; let's say, sign of $\rho$ is always smaller than 0, then, which direction do I need to take with one-tailed test? I'm asking this because t-distribution is not symmetric around 0.
In case I don't have a background knowledge, sign of $\rho$, then, what is the correct way to design H0 & H1? to me, two tailed one is the most tempting one because I don't need to care about sign, but it doesn't give me any information about correlation or anti-correlation.
Thanks in advance.