$X_1, X_2, ..., X_{15}$ are independently to each other and follow $N (7, 3^2)$
what distribution the following statistics follow
$T = \frac{(\bar{X}− 7)}{\sqrt{s^2/15}}$
i know this follow t distribution $t_(n-1) =t_{14}$
but how do i find what distribution $T^2$ follows, can i just multiply it?
$T = (\frac{(\bar{X}− 7)}{\sqrt{s^2/15}})^2=\frac{Z^2*(n-1)}{\chi_{(n-1)}}$
In general if $T\sim t(n)$ ($t$ distribution with $n$ degrees of freedom), then $T^2\sim F(1, n)$ (i.e. follows an F-distribution). To see this since $X\sim t(n)$ it follows that we can write (in distribution) that $$ T=\frac{Z}{\sqrt{W/n}} $$ where $Z$ and $W$ are independent, $Z\sim N(0,1)$ and $W\sim \chi ^2_{(n)}$. It follows that $$ T^2=\frac{Z^2/1}{W/n} $$ But $Z^2\sim \chi^2_{(1)}$ so $T^2\sim F(1,n)$ by definition of the F-distribution.
In your specific case $T^2\sim F(1,14)$.