In this content unit we make the critical switch from "population mean" (or means) as the parameter of interest to inference around the population variance. Please think about examples where the parameter of interest would be the variance/standard deviation rather than the mean. (Remember that the underlying random variable/s must be normally distributed.)
An example that comes to mind for me is models that "stress test." I think about the Army Corps of Engineers which builds infrastructure and then must be sure the infrastructure is robust enough. Think about the case of a dam built by the Army Corps. Assume that the rainfaill/water supply for the dam is normally distributed. To consider "how much water" the dam should be able to accommodate, the Army Corp must understand the variance (or more importantly the std deviation). For example, "stress test" models might test whether the dam can accommodate water levels 4,5 or 6 (etc) standard deviations about the mean.
In the very tragic case of New Orleans and the water levees there, I would be interested to know how those levees were stress tested and how much "above the mean" of water supply the levees were built to accommodate and how "unusual" the event of Katrina actually was.
Another common example with variance ("volatility") is returns in stock/financial markets.
Please add and discuss your own example.