Consider two data series, $X = (x_1, x_2, \dots, x_n)$ and $Y = (y_1, y_2, \dots, y_n),$ both with mean zero. We use linear regression (ordinary least squares) to regress $Y$ against $X$ (without fitting any intercept), as in $Y = aX + \epsilon$ where $\epsilon$ denotes a series of error terms.
It can be shown that $a=\frac{\rho_{XY} \sigma_Y}{\sigma_X}$.
Suppose that $\rho_{XY} = 0.01$. Is the resulting value of $a$ statistically significantly different from $0$ at the $95\%$ level if:
i. $n=10^2$
ii. $n = 10^3$
iii. $n = 10^4$
I know that I need to find a $p$-value for each of these, and I assume that it will be in turns of $\rho$ and $n$. However, everything I attempt leaves a lingering standard deviation somewhere from the definition of $a$. How do you test significance of a least squares in this case?