# testing correlation coefficient in a bivariate normal distribution

How can I show that $\dfrac{\hat{\rho } \sqrt{N-2}}{\sqrt{1-\hat{\rho}^2}}$ has a t-student distribution with $N-2$ degrees of freedom.

I think I have to write it as a quotient of a normal $(0,1)$ and the squared root of a chi-squared distribution divided in its df. But I've been trying and could not do it.

• I think we'll need to assume that is the distribution when $\hat\rho$ is the sample correlation in a multivariate normal sample in which the two components are independent. If they are dependent the the expected value of $\hat \rho$ would not be $0$. ${}\qquad{}$ – Michael Hardy Nov 18 '15 at 5:39
• Interesting question .... remind me to look at this tomorrow..... ${}\qquad{}$ – Michael Hardy Nov 18 '15 at 5:50