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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).

Engineer_2018
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  • If the population has a Gaussian distribution and the variance $\sigma^2$ is known, then the sample mean has a Gaussian distribution with variance $\frac{\sigma^2}{n}$ no matter how small the sample size. The $t$-distribution covers the case where you are estimating the variance from the sample. In your sentence "Generally ...", the use of or is not correct. The answer to your final sentence is "no" – Henry Apr 25 '19 at 09:39
  • Thank you for the answer. I am not assuming the population is Gaussian to begin with. It could be anything. In response to your answer that : "The answer to your final sentence is "no" " , I would like to know what is the best we can assume about the sampling distribution in this case. In other words , what is the sampling distribution if sample size is less than 30. – Engineer_2018 Apr 25 '19 at 12:09
  • The $t$-distribution was specifically designed to deal with unknown variance estimated from the observation. Its effect is to widen confidence intervals (reduce $p$ values if you prefer), especially for small sample sizes, making it conservative; some people you (not me) might think this conservatism could be helpful (albeit theoretically unjustifiable) if you are using the Central Limit Theorem to argue that the sample mean has an approximately normal distribution – Henry Apr 25 '19 at 13:31
  • Sorry, I did not understand your last statement starting with 'some people you.... – Engineer_2018 Apr 25 '19 at 13:44
  • Simple rule: If data are normal and population SD $\sigma$ is unknown and estimated by sample SD $S,$ always use t test, regardless of sample size. (In this case a z-test is never theoretically correct, but if $n > 30,$ A z-test may give approx. correct answ. But why mess with this when t is easy and correct? A z-test amounts to assuming $S$ to be a perfect estimate of $\sigma.)$ – BruceET Apr 25 '19 at 18:30
  • Thank you.@BruceET will your statement be correct if data is not normal – Engineer_2018 Apr 26 '19 at 08:34

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