Let $X_{i}$ be $N(\mu, \sigma^2)$
I want to :
(1) Prove that $\left\{ (X_{1}-\bar{X})/S, (X_{2}-\bar{X})/S,\cdots,(X_{n}-\bar{X})/S\right\}$= $T$ does not depend on S
(2) find pdf of $(X_{1}-\bar{X})/S$.
I'm guessing independence follows from the Basu's theorem.
My attempt: ($\bar{X}$, $S^2$) is a complete sufficient statistic for $(\mu, \sigma^2)$. That means if I prove that the distribution of $T$ does not depend on $(\mu, \sigma^2)$ the proof will be complete. I was able to present $(X_{1}-\bar{X})/S$ as $(Z_{1}-\bar{Z})/S$ (the same form but in terms of Z from $N(0, 1)$ ), but don't now what to do next.
So I'm stuck on the ancillary statistic in (1) and have no idea how to deal with (2)
I'd be glad if someone could help me
Edit: Basu's Theorem: If $T$ is a complete sufficient statistic for $θ$ , and A is ancillary to $θ$ then $T$ is independent of $A$