# Test statistic for regression slope in terms of correlation

Consider two data series, $X = (x_1, x_2, ..., x_n)$ and $Y = (y_1, y_2, ..., y_n)$, both with mean zero. We use linear regression (ordinary least squares) to regress $Y$ against $X$ (without fitting any intercept), as in $Y = aX + ε$ where $ε$ denotes a series of error terms. Suppose $\rho_{XY}$ is the sample correlation.

Question: Derive a test statistic to test the significance of the regression slope $a$ in terms of the sample correlation $\rho_{XY}$ (Assuming that the sample size is sufficiently large such that this statistic is normal).