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It seems much more logical to me to teach confidence intervals for large samples first, when the CLT (just taught) implies that that the sample means are normally distributed, no matter the distribution, no matter whether the standard deviation is known. Is there a reason not to do it this way? Why do textbooks start with the case of normally distributed, standard deviation known?

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    $\begingroup$ One evident problem is that the CLT does not imply the things you state unless you take special care in defining what a "large sample" is and are clear about assuming the standard deviation exists. That can sow confusion, especially among neophytes, and introduces ideas that (although mathematically interesting) seem likely to distract from the concept of a confidence interval. $\endgroup$ – whuber Oct 22 '17 at 17:04
  • $\begingroup$ Starting with a sample that somehow is normally distributed with standard deviation known also is likely to sow confusion. In any case, would you recommend following the textbook order: 1) CLT, 2) CI for any size sample, normally distributed, $\sigma$ known, 3) large sample, any distribution, $\sigma$ unknown, 4) small sample, normally distributed, $\sigma$ unknown? I know students find this difficult, as there's a lot of jumping around. Perhaps it's unavoidable. Or perhaps cover 2) and 4) and then 1) (CLT) and 3)? $\endgroup$ – jtr13 Oct 22 '17 at 17:53

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