How to motivate undergrads to take an Intro to Math Stats course How would you motivate an undergraduate student to take an Introduction to Mathematical Statistics course, if they know that much of their time will be spent proving results they've already seen in Introduction to Applied Statistics? That is, much (though certainly not all!) of the course is "just" about proving mathematically that standard methods for p-values and confidence intervals (z and t tests and intervals; F and chi-square tests) work as advertised.
Context: I teach at a US small liberal arts college. Here is the course description for our version of this class:

Topics in Statistical Inference Building on their background in probability theory, students explore inferential methods in statistics and learn how to evaluate different estimation techniques and hypothesis-testing methods. Students learn techniques for modeling the response of a continuous random variable using information from several variables using regression modeling. Topics include maximum likelihood and other methods estimation, sample properties of estimators, including sufficiency, consistency, and relative efficiency, Rao-Blackwell theorem, tests of hypotheses, confidence, and resampling techniques.

We have used textbooks such as Hogg, McKean, & Craig (table of contents) or Wackerly, Mendenhall, & Scheaffer (table of contents).
Our statistics majors are required to take Math Stats, but I try to give better motivation than just "it's required." And occasionally other students ask me why they should consider taking this course too. What are some ways I could convince them to see the value of taking Math Stats?
I'll post my own answer with what I currently tell them, but I expect that the community here has other great suggestions.

If the undergraduates have just taken a course in Introduction to Applied Statistics, well, frankly I can't blame them for not taking the Introduction to Mathematical Statistics course. For interested students, I would think that the "theory" behind applied statistics would be at least minimally presented in the applied statistics course; enough for them to understand it in principle. And these things take a while, and some experience, to intuitively understand and be cogitated upon. Unless perhaps there was some deficiency in the first course in explaining basic concepts and they got stuck on that.
I would suggest that perhaps other courses might fit students' progression up the statistical ladder, such as regression (logistic, Poisson and others); sampling methods; a software-oriented epidemiology course, or even an applied course in genetics.  These would help them to intuit statistical logic. It seems your college might be forcibly dividing intro statistics in to an "applied" camp where the basic how-to is performed, and a "mathematical" camp where more mathematical, probability-based and inter-related statistics are looked at under the hood.    You could always change the name of the course to make it a more logical follow-on to the first course.  My guess is that those who took the first course are actually hungry for more applications of statistics, and this second course might be better to have some real-world examples, making it "semi-applied."
With that said, I appreciate trying to teach the undergraduates more about the logic of inference and probability and so forth, beyond what they learned in the nuts-and-bolts course.  But it might be (or sound) overwhelming for them.  In my own personal journey I've come to appreciate statistical theory on things well after I've had multiple statistical courses on everything from design of experiments to logistic regression and probability, and a lot of real-world research.  And I've found the theory to usually be much simpler than it sounds; again, partly because a certain amount of applied statistics has been put under my belt.
