Testing Correlation coefficient

Let $(X_i, Y_i), i=1,2,\cdots, n$ be a random sample of $n$ pairs of observations drawn from a bivariate normal population with correlation coeffficient $\rho$.

The hypothesis problem can be setup as

Situation Hypothesis Testing Problem
Situation A $H_0: \rho=0$ against $H_a : \rho < 0$ (Left-tailed)
Situation B $H_0: \rho=0$ against $H_a : \rho > 0$ (Right-tailed)
Situation C $H_0: \rho=0$ against $H_a : \rho \neq 0$ (Two-tailed)


The test statistic under $H_0: \rho=0$ is

$t = \dfrac{r}{\sqrt{1-r^2}}\sqrt{n-2}$


  • $r$ is the sample correlation coefficient between $X$ and $Y$,
  • $n$ is the number of pair of sample observations,

The test statistic $t$ follows $t$ distribution with $n-2$ degrees of freedom.

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