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) |
Formula
The test statistic under $H_0: \rho=0$ is
$t = \dfrac{r}{\sqrt{1-r^2}}\sqrt{n-2}$
where,
$r$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.
