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While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in?

I am asking this in the context of basic statistical inference, when one learns how to construct confidence intervals for $Z =\frac{\overline{X}-\mu}{\sigma/\sqrt{n}}$ and then moves on to the case where $\sigma$ is unknown... so that $T=\frac{\overline{X}-\mu}{s/\sqrt{n}}$ is used instead.

For the normal distribution, you can show many real-life examples. Its historical forthcoming is also natural. As for the Student's, its history is actually quite revolutionary, and adjustments from the original work of Gosset were also involved.

Any suggestions, ideas would be greatly appreciated!

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At that level, if it is taught at all, I think that unfortunately it has to be taught as a black box rule. An easy to remember "explanation" should be given, that uncertainty about the true variance forces a broader confidence interval. – André Nicolas Feb 5 '13 at 17:13

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