This is from an assignment.
The problem:
Given $n=15$ and assuming $Z_1, ..., Z_n$ are iid $N(\mu, \sigma^2)$, let $\bar{Z}$ be the sample mean and $S^2 = \frac{1}{n-1} \sum^n_{i=1} (Z_i - \bar{Z})^2$, define the following random variable $$ V = \frac{(Z_1 - \mu)^2}{\sigma^2} $$ and calculate $$ P(V > 1.801) $$
My solution:
The second part is not an issue, I am a little stuck on what distribution $V$ follows. I know that $(Z - \mu)$ subtracts the mean from the normal distribution and gives $\mu = 0$, and I know that $Z^2$ gives a $\chi^2$ distribution. I think that dividing by $\sigma^2$ probably standardizes the variance to produce a standard normal $\chi^2$ if that exists. To my mind though, that would look like $$ \left( \frac{Z_1 - \mu}{\sigma^2} \right)^2 $$ to first create the standard normal then square to produce a $\chi^2$ distribution.
Any prods in the right direction would be appreciated.
Edit: Some of the above information may be superfluous and for different parts of the question which I have not asked about.