## Testing Homogeneity of Two 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 population with correlation coeffficient $\rho$.

Let $r$ be the observed correlation coefficient between $X$ and $Y$.

We wish to test the hypothesis $H_0 : \rho =\rho_0$ against $H_a : \rho \neq \rho_0$.

## Formula

The test statistic for testing $H_0$ is
```
$$
\begin{aligned}
Z&=\dfrac{U-\xi}{\sqrt{\frac{1}{n-3}}}
\end{aligned}
$$
```

where
```
$$
\begin{aligned}
U&=\frac{1}{2}\log_e \bigg(\frac{1+r}{1-r}\bigg)
\end{aligned}
$$
```

and
```
$$
\begin{aligned}
\xi & =\frac{1}{2}\log_e \bigg(\frac{1+\rho_0}{1-\rho_0}\bigg)
\end{aligned}
$$
```

The test statistic $Z$ follows Standard normal distribution.