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We all heard about the simplification to use a t-distribution when sample size is small (n<30) and Z-distribution when n is large (n>30); indeed, with increasing n, the distribution of the sample means will converge to the normal distribution thanks to the CLT. But what truly happens when n is not large enough? The requirements for a Z-distribution are: 1) The sample mean is normally distributed AND 2) The population standard deviation σ is known (in order to use σ for our statistic test) The requirements for a t-distribution are: 1) The sample size is large (n>30) OR 2) The sample is normally distributed (may require separate normality tests) OR 3) The population is known to be normally distributed but with standard deviation σ unknown (so we must use s instead). So really, when the sample size is small (n<30), AND when the population from which the sample is taken is known to be normally distributed with... read more

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