A clinical trial evaluates a drug's effect on heart rate, where the improvement follows a normal distribution $\Delta \sim N(\mu, \sigma^2/n)$. If the population variance $\sigma^2$ is unknown, which distribution should the test statistic $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ follow under the null hypothesis, and why is it distinct from the $z$-test?