Suppose 100 samples of size $\displaystyle n=3$ are taken from each of the pdf's

(1) $\displaystyle f_Y(y)=2y$, $\displaystyle 0\le{y}\le{1}$
and
(2) $\displaystyle f_Y(y)=4y^3$, $\displaystyle 0\le{y}\le{1}$

and for each set of three observations the ratio

$\displaystyle \frac{\bar{y}-\mu}{s/\sqrt{3}}$

is calculated, where $\displaystyle \mu$ is the expected value of the particular pdf being sampled. How would you expect the distributions of the two sets of ratio to be different? How would they be similar?


I know that the t-distribution is robust, so that it should be fairly unaffected by the underlying distributions unless they are extremely skewed and/or n is small. Clearly a sample of size three is fairly small, but how do I answer this question? I do not have software to automatically sample from these distributions to form a histogram and superimpose a t-distribution on top of it. Is there an easier way?