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The Pearson product-moment correlation coefficient is a measure of the linear relationship between two variables $X$ and $Y$, giving a value between +1 and −1.
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When testing Pearson's r, when should I use r-to-t transformation instead of [Fisher's] r-to...
At least two tests are common for testing the significance of a Pearson correlation coefficient.
Comparing $t=\frac{\sqrt{r^2/(1-r^2)}}{\sqrt{\frac{1}{N-2}}}$ to Student's $t$-distribution with $N-2 …
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When testing Pearson's r, when should I use r-to-t transformation instead of [Fisher's] r-to...
That is correct.
(The answer to my question was posted by another user as a since-deleted comment. I'm adding this comment so I can mark the question as answered.)