By definition, the T distribution is the ratio of standard normal variable and sqrt of scaled $\chi^2$ variable. The "popularized" version of (one sample) t statistic goes like this: $\frac{\bar{x}-\mu}{S/\sqrt{n}}$. But really, this is derived using $t = \frac{Z}{\sqrt{\frac{\chi^2_{(n-1)}}{n-1}}} = \frac{\frac{\bar{x}-\mu}{\sigma/\sqrt{n}}}{\sqrt{\frac{(n-1)S^2/\sigma^2}{n-1}}} = \frac{\bar{x}-\mu}{S/\sqrt{n}}$.
The $Z$ comes from a CLT assumption that sample mean approximates normal distribution. Informally, this only holds for the magic number $n >= 30$. Since the t statistic is constructed based on this assumption, then shouldn't one-sample t-test only be applied when $n >= 30$?
