Is there any commonly used method to test the zero correlation between $X$ and $Y$ using the sample correlation coeffcient from two samples $\{x_k\}_{k=1}^n$ and $\{y_k\}_{k=1}^n$?
In another word, if $\boldsymbol{x}=(x_1,\dots,x_n)^\top$ and $\boldsymbol{y}=(y_1,\dots,y_n)^\top$ are two standardized vectors, i.e., $\sum_{i=1}^nx_i=0$ and $\sqrt{\sum_i x_i^2}=1$, is there any method to numerically judge $\boldsymbol{x}\perp \boldsymbol{y}$?
Actually $\boldsymbol{x}$ and $\boldsymbol{y}$ are two observed noisy data vectors, so I do not think it is appropriate to use $\boldsymbol{x}^\top \boldsymbol{y}=0$ as the criterion.