Significant Others

Suppose that we have a set of data $\{(x_1,y_1),\dots, (x_n,y_n)\}$, where both $x_i$ and $y_i$ are real-valued and mean $0$. We fit a linear regression line without intercept to regress $y$ onto $x$. This yields a line $y = \hat{\beta}_0 x + \varepsilon$, $\varepsilon$ being an error term. Given that Corr$(X,Y) = \dfrac{1}{10\sqrt{3}}$, find the smallest value of $n$ so that the results of the hypothesis test $H_0: \beta_0 = 0$ vs. $H_A: \beta_0 > 0$ are significant at the $95\%$ level. You may use the fact that the $95$ and $97.5$ percentile values are approximately $1.645$ and $1.96$.