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.
 A: 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.
A: *

*Do you hope to do research on statistical or machine-learning methods? Even if this class just proves conclusions that you've already seen, the mathematical techniques and frameworks behind these proofs are the same ones used by research statisticians to develop new cutting-edge results. This class is the first step towards making your own contributions to statistical methodology. It will also help you sanity-check a colleague's work when they say they've developed a "better" statistical method for some scenario. (How is it better: more efficient? more robust? etc. Does it actually work the way they think it does?)

*Do you appreciate beauty in math? I find that some of the math techniques or results themselves are beautiful. For instance, you can approach several common situations from totally different angles (MLE, MOM, Bayesian posterior mode, minimizing MSE...) and they all give the same answer (sample mean)! How cool is that?

*Do you only care about applications, not the "pure" math itself? Even so, a Math Stats course will give you a much deeper understanding of what these methods are doing. Personally, I had seen p-values and confidence intervals many times, but they did not really click for me until I took Math Stats. It will also help you learn to watch out for situations when the standard methods won't work and possibly develop your own solutions or workarounds.

*Do you want to work with data-minded colleagues but not analyze data yourself? Even so, you'll be better prepared to follow other people's new developments in statistics, AI, and machine learning, so you can sensibly decide whether they're useful to your work. For instance, differential privacy has been a recent hot topic in my areas of work, but it's not trivial to understand what it's even trying to solve or how it proposes to do so; some Math Stat background definitely helps.

A: 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.
