Can the t-test be used to test the difference between percentiles of 2 samples? For example, instead of testing the difference in mean, I want to test the difference in the 75th percentile of the 2 groups.
Does the central limit theorem hold? and what would be the equation for the standard error?
 A: Quoting from Theorem 7.5.1 (p243) of Bain & Englehardt (1992), except for notation:
Let $X_1, X_2, \dots, X_n$ be a random sample from a continuous distribution with PDF $f(x)$ that is continuous and nonzero at the $p$th percentile $x_p,$ for $0 < p < 1.$ If $k/n \rightarrow p$
(with $k-np$ bounded), then the $k$th order statistic $X_{(k)}$ is asymptotically normal with mean $x_p$ and variance $c^2/n,$
where $$c = p(1-p)/f(x_p).$$
So there is a CLT for the 75th percentile and the asymptotic
variance is as specified in the theorem. Instead of requiring
a finite population variance $\sigma^2,$ as in the CLT for
means, the requirement is roughly that the quantile $x_p$ of the distribution be precisely defined, with $f(x_p) > 0.$
Suppose you have a sample of size $n=625$ from a population distributed as $\mathsf{Norm}(\mu = 100, \sigma = 15),$ with 75th
percentile $110.1173$ and $$c^2/n = \frac{3/16}{0.0212n} = 0.6684.$$
f = dnorm(qnorm(.75, 100,15), 100,15); f
[1] 0.0211851
(3/16)/(625*f^2) 
[1] 0.6684363

If we simulate $m=100\,000$ such samples, we see that the variance
of the resulting $m$ 75th quantiles is in good agreement with the theoretical
asymptotic variance.
q = replicate(10^5, as.numeric(
     quantile(rnorm(625, 100,15),.75) ))   
var(q)
[1] 0.6679301

I am not sure exactly what null and alternative hypotheses
you will test and for what distribution. Of course, the 2-sample t test programmed into statistical software programs uses means and variances. I suppose power will be poorer using 75th percentiles than using means. Notice that the variance of a sample mean of $n=625$ observations from the normal distribution of my example above has variance $\sigma^2/n = 15^2/625 = 0.360 < 0.668.$ 
So I will leave
the rest to you.
