As I understand it, Student's t will asymptotically approximate the Z statistic at sample sizes approaching infinity (see the infinite degrees of freedom row for degrees of freedom in most introductory statistics books). However, Z tests are applicable when populations are assessed in total.
Thus, when the population size is finite, known, and relatively small, it seems that a t-statistic may be overly conservative. Specifically, if the population is 50 and you sampled 49, it seems that the appropriate critical value would be closer to the associated critical value for Z (given the alpha criterion) than it would be to t (given the alpha and degrees of freedom).
Is it true that t is conservative in these small population cases? If so, is there a correction to the t critical values to account for cases where the population is small?
Best Answer
When you have a finite population and the sample size is more than 5-10% of the population then you should use the Finite Population Correction, that is you multiply your standard error times $\sqrt{\frac{N-n}{N-1}}$, see this link.
You should still use the t-distribution, but the standard error will be smaller to account for the large portion of the population in the sample.