I am asking a follow up question to this question. Why prefer the t-score when the sample size is low?
I have seen mathematical proofs that in variance unknown case, the t-statistic follows a t-distribution regardless of the choice of sample size. This is usually proved from first principles by observing that t statistic is ratio of a Normal and Chi-Square random variables.
Generally I have seen recommendation to use t-tests in low sample size case and/or variance unknown.
I want to know whether we can prove that t-statistic follows a t-distribution when the population variance is known(i.e not estimated from samples) but the sample size is small (< 30).