To determine if there exists a [[Linear Regression|linear]] [[Correlation|correlation]] between two [[Normal Distribution|normally distributed]] [[Random Variable|random variables]], setup a two-tailed [[Hypothesis Test|hypothesis test]] on the population correlation coefficient $\rho$ with test [[Statistic|statistic]] as the [[Pearson Correlation Coefficient|Pearson correlation coefficient]] $r$. Establish the [[Null Hypothesis|null hypothesis]] that there exists no [[Correlation|correlation]]. $ \begin{align*} &H_0: \rho = 0 \\ &H_1: \rho \ne 0 \end{align*} $ Use the [[Z-Score|standardised]] test statistic $t = \frac{r}{\sqrt{1 - r^2}/\sqrt{n - 2}}$ (using the [[Variance|variance]]/[[Standard Deviation|standard deviation]] of the correlation coefficient) on the [[t-distribution]] with $(n - 2)$ degrees of freedom (lose 2 degrees from two variables). Reject $H_0$ if $t \ge t_{\alpha / 2}(n - 2)$ or $t \le -t_{\alpha / 2}(n - 2)$. If the null hypothesis is rejected, then there exists a linear relationship between the two variables.