  We show that the statistics of the continued fraction expansion of a randomly
chosen rational in the unit interval, with a fixed large denominator $q$,
approaches the Gauss-Kuzmin statistics with polynomial rate in $q$. This
improves on previous results giving the convergence without rate. As an
application of this effective rate of convergence, we show that the statistics
of a randomly chosen rational in the unit interval, with a fixed large
denominator $q$ and prime numerator, also approaches the Gauss-Kuzmin
statistics. Our results are obtained as applications of improved non-escape of
mass and equidistribution statements for the geodesic flow on the space
$SL_2(\mathbb{R})/SL_2(\mathbb{Z})$.
