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

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Yes, because the slope can be expressed as:

$$ \beta = \frac{cov(X, Y)}{var(X)} $$

While the correlation coefficient is:

$$ \rho= \frac{cov(X, Y)}{\sqrt{var(X)*var(Y)}} $$

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.

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