The hardest thing to teach in any introductory statistics course is the sampling distribution of the sample mean, a topic that is at the center of the typical intro-stat-class-for-nonmajors. All of probability theory builds up to it, and then this sample mean is used over and over again for inferences for averages, paired and unparied differences, and regression. This is the standard sequence, as in the books by Moore and McCabe, and De Veaux et al.
The trouble is, most students don’t understand it. I’m not talking about proving the law of large numbers or central limit theorem–these classes barely use algebra and certainly don’t attempt rigorous proofs. No, I’m talking about tha dervations that lead to the sample mean of an average of independent, identical measurments having a distribution with mean equal to the population mean, and sd equal to the sd of an individual measurement, divided by the square root of n.
This is key, but students typically don’t understand the derivation, don’t see the point of the result, and can’t understand it when it gets applied to examples.
What to do about this? I’ve tried teaching it really carefully, devoting more time to it, etc.–nothing works. So here’s my proposed solution: de-emphasize it. I’ll still teach the samling distribution of the sample mean, but now just as one of many topics, rather than the central topic of the course. In particular, I will not treat statistical inference for averages, differences, etc., as special cases or applications of the general idea of the sampling distribution of the sample mean. Instead, I’ll teach each inferential topic on its own, with its own formula and derivation. Of course, they mostly won’t follow the derivations, but then at least if they’re stuck on one of them, it won’t muck up their understanding of everything else.