> Alice should sample the population at random, take the height measurements, calculate the average, plot the distribution, calculate the variance A student-t

> Alice should sample the population at random, take the height measurements, calculate the average, plot the distribution, calculate the variance A student-t test is basically a lossy encoding of that information you describe (the sample mean, the variance, and the sample size). > If the distribution is not sufficiently smooth then continue to take measurements until it's smooth, or unchanging. From a frequentist standpoint, this would be considered sloppy and bad methodology. You don't keep sampling until you get the results you want (or until the null hypothesis is invalidated). That said, as I mention in a comment above, the fact that you can keep sampling until the null hypothesis is invalidated (and that it is mathematically always guaranteed to happen eventually) is a big problem with the concept of hypothesis testing in the first place.