I need to calculate sample sizes, but I'm curious as to why the commonly used Krejcie-Morgan tables use a chi-squared formula rather than a student-t or a normal distribution. I see quite a few other papers that use the latter distributions instead. Is this related to an assumption about the population following a normal distribution?
After searching a while, seems that the formula is coming from constructing the confidence interval for proportion in a small sample, using hypergeometric distribution to model the observations, and using normal approximations for the hypergeometric distribution as well. Thats why you come up the square of normal quantile (which is the chi-square quantile).