I've been asked to contribute some lectures (or parts of lectures) to a course on "Mathematical Modelling", from a statistical perspective. This is to a rather mixed group of Mathematics undergraduates: some may have only seen one course on probability and statistics before.
There are a huge number of books with titles like "Introduction to Mathematical Modelling" (first example from a search) which typically concentrate upon the use of e.g. differential equations. Such sources often have a little on what I would call Monte Carlo simulation: adding some randomness to a model, and then running simulations. However, at least in the books I have looked at, there is extremely little which is "data driven".
- I want to stress that such books are mathematically fairly unsophisticated, but at the same time, could not be called "popular science".
What I'm looking for is little case studies which start with some data, then discuss various probability models, then fit those models, and then make predictions or inferences, undertake some hypothesis testing (well, I much prefer a Bayesian point of view, but this is not how our students are taught) etc. However, what I've found is:
- Any book with "statistical modelling" in the title seems much too advanced: e.g. starting off with generalised linear models
- I like the approach taken by many Bayesian textbooks (to pick an example, Sivia and Skilling) but these tend still be rather sophisticated (relatively speaking). I guess I'm really after presentations in this style, but which assume less of the reader, without going so far as to be "popular science".
- Of course, elementary stats textbooks have plenty of examples, but these tend to motivate things backwards from what I'm looking for: e.g. after introducing the Poisson distribution, there is some data presented (maybe just the sample mean given in fact), and some comments. But missing is seemingly why we might choose the Poisson distribution over other choices, etc.
I'd love some example sources which are in the style I'm after.
(I should also say that I'm hoping to escape actually giving lectures, but rather, give some sources to a colleague...)