## 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.