I am going to teach statistics as a teaching assistant for the second half of this semester to CS-oriented undergraduate students. Most of the students took the class has no incentive to learn the subject and only took it for major requirements. I want to make the subject interesting and useful, not just a class they learn to get a B+ to pass.
As a pure-math PhD student I knew little on the real-life applied side. I want to ask for some real-life applications of undergraduate statistics. Examples I am looking for are ones (in spirit) like:
1) Showing central limit theorem is useful for certain large sample data.
2) Provide a counter-example that central limit theorem is not applicable (say, the ones following Cauchy distribution).
3) Showing how hypothesis testing works in famous real life examples using Z-test, t-test or something.
4) Showing how overfitting or wrong initial hypothesis could give to wrong results.
5) Showing how p-value and confidence interval worked in (well known) real life cases and where they do not work so well.
6) Similarly type I, type II errors, statistical power, rejection level $\alpha$, etc.
My trouble is that while I do have many examples on probability side (coin toss, dice toss, gambler's ruin, martingales, random walk, three prisoner's paradox, monty hall problem, probability methods in algorithm design, etc), I do not know as many canonical examples on the statistics side. What I mean is serious, interesting examples that has some pedagogical value, and it is not extremely artificially made up that seems very detached from real life. I do not want to give students the false impression that Z-test and t-test is everything. But because of my pure math background I do not know enough examples to make the class interesting and useful to them. So I am looking for some help.
My student's level is around calculus I and calculus II. They cannot even show the standard normal's variance is 1 by definition as they do not know how to evaluate the Gaussian kernel. So anything slightly theoretical or hands-on computational (like hypergeometric distribution, arcsin law in 1D random walk) is not going to work. I want to show some examples that they can understand not just "how", but also "why". Otherwise I am not sure if I will be proving what I said by intimidation.