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I know that the test for significance of linear correlation is:

$t=\frac{\sqrt{n-2}\rho}{\sqrt{1-\rho^2}}$

with ($n-2$) freedom degrees, but i need the proof. Do you know why the test follows a Student t distribution?

Thanks!

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1 Answer 1

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The test statistic has a t-distribution because it's actually just the t-statistic for the slope coefficient in a simple linear regression recast to be in terms of the correlation.

The sampling distribution of this statistic is derived from scratch directly in terms of the correlation coefficient in numerous places - for example, Hogg & Craig (1965, Introduction to mathematical statistics, 2nd Ed p361-363), but the assumptions are the same as the linear regression case, and the statistics correspond.

If you're happy that the test of a slope coefficient in a simple regression is distributed as a $t_{n-2}$, this is effectively the same problem in a different guise.

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