We give the convergence rates of the random sample size bootstrap approximations of the distribution of the sample mean. Using the quantity $\alpha^2_n=E[(\frac{N_n}{n}-1)^2]$, where $N_n$ is the random sample size, we obtain a convergence rate $O(n^{-\frac 12}(\log\log n)^{\frac 12})$ if $\alpha_n=O(n^{-\frac 12}(\log\log n)^{\frac 12})$.