Suppose $X_1, \dots, X_n$ are i.i.d. $N(\mu, \sigma^2)$. The Wald statistic for testing $H_0: \mu = \mu_0$ is $W = n(\bar{X} - \mu_0)^2 / \hat{\sigma}^2$. If the variance $\sigma^2$ is unknown and estimated by the sample variance $S^2$, what is the exact distribution of $W$?