An excellent post, Wayne, but I don't think that last statement is quite right. If the F-test gives a probability of 0.23 for material C, I believe it's saying that there is a 23% that, given the noise level in the data, the fit would indicate that C was present when it was not. That is not the same thing as saying there is a 77% chance of C being present. 

To see this, imagine very, very noisy data. Including C in the fit might very well improve the fit in the sense of an R-factor--maybe, in fact, there's a 45% chance of a modest improvement with a given set of very noisy data, even if there's no C present. That does not mean that a result like that should lead to the conclusion that C is more likely than not present (55%).

--Scott Calvin

Sarah Lawrence College