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Suppose you are given a correlation matrix and a sample size. How do you calculate which correlation coefficients are statistically significant?

If there are $K$ variables then there are $K^{2}$ entries in the correlation matrix, of which $K$ are on the diagonal. Half of the entries are the same, so there are $\displaystyle\frac{K^{2} - K}{2}$ correlation coefficients. Assuming the Bonferroni correction, the critical value $\alpha = 0.05$ would be divided by that number.

However, how do I obtain the P values?

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You can obtain $p$-values for Pearsons correlation using Student's $t$-distribution, since

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

has $t$-distribution (see Wikipedia for learning more and further references).

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