Let's suppose I have calculated the correlation between stock prices and P/E ratios from each quarter from 2010 to 2015 (a total of 24 quarters) to be 0.85. Assuming a 5% significance level, I want to know if this correlation is statistically significant.
My process went like this: 24 periods means I have 22 degrees of freedom. My null hypothesis is that the correlation between all stock prices and P/E ratios for this stock is 0. My alternative hypothesis is that it is not 0. Therefore I believe this is called a "two-tailed test." Since this is a two-tailed test, the critical value is 2.074.
Now I can use the test statistic $t = \frac{r\sqrt{n - 2}}{\sqrt{1-r^2}}$
where $r$ is the correlation coefficient and $n - 2$ is the degrees of freedom. Plugging in the values aboe, I get that $t = \frac{0.85\sqrt{22}}{\sqrt{1-(0.85)^2}}$. This equals $7.568$, which is greater than my critical value of $2.074$. Therefore this correlation is statistically significant.
Is my process correct?