Suppose $X$ and $Y$ are nonnegative random variables such that $\mathrm{Pr}(X\geq t)\leq\mathrm{Pr}(Y\geq t)$ for all $t\geq0$. Now take $X_1,\ldots,X_n$ to be independent with the same distribution as $X$, and similarly for $Y_1,\ldots,Y_n$ with $Y$. I would like to know if the following is necessarily true: $$ \mathrm{Pr}\bigg(\frac{1}{n}\sum_{i=1}^nX_i\geq t\bigg)\leq\mathrm{Pr}\bigg(\frac{1}{n}\sum_{i=1}^nY_i\geq t\bigg) \qquad \forall t\geq0. $$ By assumption, it's true for $n=1$. For large $n$, the sample averages will approach $\mathbb{E}[X]$ and $\mathbb{E}[Y]$, respectively, which satisfy $$ \mathbb{E}[X] =\int_0^\infty\mathrm{Pr}(X\geq t)~dt \leq\int_0^\infty\mathrm{Pr}(Y\geq t)~dt =\mathbb{E}[Y]. $$ So intuitively, the inequality of interest should also hold for large $n$. Can anything funny happen for moderately sized $n$?

(As the title suggests, I'm interested in this as a tool analyze measure concentration.)