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.

You could use the following test to check whether there is significant correlation between $X$ and $Y$. Assume that you have the observations $(x_i,y_i), i =1,\dots,n$.
The null and alternative hypothesis are given by:
$$
H_0: \, \rho = 0 \quad vs. \quad H_1: \rho \neq 0
$$
The test statistic is given by:
$$
T = \sqrt{n-2}\frac{\hat{\rho}}{\sqrt{1-\hat{\rho}^2}}\overset{H_0}{\sim} t_{n-2}
$$
Where $\hat{\rho}$ is the sample estimate for the correlation coefficient, i.e.
$$
\hat{\rho}=\frac{\frac{1}{n}\sum_{i=1}^n((x_i-\bar{x})(y_i-\bar{y}))}{\sqrt{\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})^2} \cdot \sqrt{\frac{1}{n}\sum_{i=1}^n(y_i-\bar{y})^2} }
$$
Thus, the null is rejected if $\vert T\vert >t_{n-2;1-\frac{\alpha}{2}}$.
