How do we know if the correlation is significant? Suppose that we have continuous data $(X_1,Y_1),\dots,(X_n,Y_n)$. Suppose that $r_{x,y}$ is the Karl-Pearson correlation coefficient between $X_i$'s and $Y_i$'s. For what range of values of $r_{x,y}$, can we really decide that there may indeed be a linear relationship between $X_i$'s and $Y_i$' and proceed to predict $Y$ by using a linear regression?
I'm sure the topic concerning this question should be a well-studied one. I did a little search here; couldn't find relevant posts. Any answers to the above question or pointers to such a study is greatly appreciated.

All you need is to compute the degree of freedom of the system, which is the number of participants minus $2$, and then refer to the table of critical values for $r$, which can be found here.
However, this gives you nothing but the mathematical significance, you still need to have a look at the scatterplot to see if a linear relationship really is a good guess or not.
