The t-test for a Pearson correlation

In short

• Test statistic: $t = r\sqrt{n-2}/\sqrt{1-r^2}$, t-distributed with n − 2 degrees of freedom.
• It is a parametric test: both variables are assumed to be normally distributed.
• The sign of t is the sign of r; t grows as |r| rises and as n rises.
• Two-sided test: reject if t is beyond $\pm t_{\alpha/2}$. One-sided test: put all of $\alpha$ in the tail named by $H_a$.
• Rejecting $H_0: \rho = 0$ means the evidence is sufficient to say the correlation differs from zero; failing to reject does not prove it is zero.