# Sampling theory of correlation; sign of population correlation coefficient & hypothesis design

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.

1. Does the test statistic above give me any information about sign of $$\rho$$?

2. 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.

3. 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.