# Is the significance of the slope equivalent to the significance of the correlation?

Is the significance of slope for linear regression the same thing as the significance of the correlation?

$$\beta = \frac{cov(X, Y)}{var(X)}$$
$$\rho= \frac{cov(X, Y)}{\sqrt{var(X)*var(Y)}}$$
Whatever significance test you do can be justified with an exact permutation test that shuffles the dependence of X on Y, which leaves $$var(X)$$ and $$var(Y)$$ as constants, so significance tests on both the slope and the correlation coefficient can be imagined as a significance test on the covariance of your two variables.