Z-scores and distributions

Some univariate distribution results that the reader should be familiar with: let $x_i\sim N(\mu,\sigma^2)$ for $i=1\cdots n$ then $\bar{x}=1/n\sum_ix_i\sim N(\mu,\sigma^2/n)$ and so $\frac{\bar{x}-\mu}{\sqrt{\sigma^2/n}}\sim N(0,1)$. When $\sigma^2$ is unknown, a well known unbiased estimate is: $\hat{\sigma}^2=1/(n-1)\sum_i(x_i -\bar{x})^2 \sim \chi^2(n-1)$ and now $\frac{\bar{x}-\mu}{\sqrt{\hat{\sigma}^2}/n}\sim t(n-1)$.