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.

This is not an answer, it's a story -
My experience matriculating from an agricultural-based community college to a large urban liberal arts college was culture shock on every front. I knew I was hungry for mathematics and, especially, statistics. In 3rd year mathematics, students are exposed to proofs, like Rudin's Analysis. It doesn't feel "gentle" - but imagine being dropped into Functional Analysis or Measure Theory without a proofs background!
In stats, I often find there's no 3rd year statistics book or course that establishes fundamental statistics. I wrote a cotaught undergraduate (4th level)/graduate level probability and statistics course from Hogg and Craig as a 3rd year student.
Hogg and Craig's book perfectly reflects the awkwardness of this discipline to establish itself as an insular and self-sufficing degree of study. The introduction, for instance, discusses the probability space as a non-negative measure on a collection of Borel sets. I remember needing wikipedia to read the first pages. Not only is that definition highly sophisticated (and I think there are more general probability spaces than are described), that result is not used anywhere else in the book except that very general results about integration are used with reckless abandon. The rest of the book is multivariable calculus. Hogg and Craig provide different treatments for continuous and discrete DFs as a result. Hogg and Craig is not a good 3rd year book at all.
I think Casella Berger is a better book for this reason. It's very focused on problems and concepts. I also think students would benefit having prior or concurrent exposure to Sheldon Ross' Probability Models which goes over probability in great detail, skipping the statistical results - the difference between these fields is as underappreciated by students as it is by us practicing professionals. Lastly, I haven't found any textbook to be of adequate quality dealing with the topic of regression, definitely the most widespread and discussed area of statistics and inference- Nachtsheim et al. is perhaps accessible at the undergraduate level, but something like a capstone project would be a better way to expose students to using the software, doing analyses, and presenting the results.
How do you motivate students to study these areas? Well, the students have to motivate themselves. You could try the "carrot" method about all the opportunities they have, and you can say that it fosters "critical and quantitative thinking", although I find training in probability theory to be more useful than statistics in this regard. You can go farther to point out that virtually every scientific discipline requires its practitioners to be critical producers of and consumers of information that is presented using statistical summaries. For students going into mathematics and statistics, you can point out that there are many exciting results just waiting for them in a 4th year course, or graduate theory - a special topics day could go over the Cantor distribution function, or you could present the Hodges' Superefficient Estimator and explain why it "breaks" the Cramer-Rao bound.
