For the cdf's of Student's $(t)$ and Thompson's $(\tau)$ distributions, upper and lower bounds are obtained in terms of the normal cdf. It is then shown that, in using the normal approximation for the cdf's of these distributions, the proportional errors are uniformly smaller than $1/n$ for all $n \geqq 8$ and 13, respectively, where $n$ is the number of degrees of freedom. Similar methods may be used to derive bounds for cdf's of similar types. Examples are given.
"Errors in Normal Approximations to the $t,\tau,$ and Similar Types of Distribution." Ann. Math. Statist. 27 (3) 780 - 789, September, 1956. https://doi.org/10.1214/aoms/1177728184