Review on approximation approach for the distribution of the studentized mean

In case of a normal population the studentized sample mean has the Student distribution with the sample size minus one as degrees of freedom. Simulation reports in the literature concluded that this distribution may be hold on as an acceptable approximation, and is often better than the classical normal approximation, in case of a general finite variance population provided moderate to large sample size. We provide a mathematical proof and add simulation evidence. A comparison of the performances of both approximations for the true distribution of the studentized statistic is given for the cumulative distribution function and for the quantile function, by asymptotic expansions and by simulations. A population condition such that the student approximation is universally better than the normal approximation is obtained.