--I reiterate to all: READ THE WEB PAGE http://rangevoting.org/CombinedTestFail.html Has anybody? This web page begins with the problem description then solves the case N=2, which everybody here seems to be having immense trouble comprehending for some mysterious to me reason, in several ways, showing the solution corresponds exactly, word for freaking word, with the problem description. Once you follow the N=2 case, and agree that (2b-a)a is the only solution, hopefully your mental block will be overcome. I evidently have triggered some kind of mass mental block, which I unfortunately am also mentally blocked from comprehending what the block is. I guess it must be my fault, but not really seeing why. Perhaps my new answer to Greathouse below overcomes/identifies the communication difficulty. The good news is, this leads to astonishingly beautiful mathematics, and I still have not completely understood it, but I'm getting a good way there, as is explained on the web page.
Ch.Gr: Let's say that we're testing a coin to see if it's unfairly biased towards heads. The null hypothesis is that it's a fair coin. Test 1 consists of flipping the coin twice and seeing if both times it comes up heads; p = 0.25. Test 2 consists of flipping it once and seeing if it comes up heads; p = 0.5. The chance that a fair coin would fail both tests is 1/8 for a p-value of 0.125. Your formula gives 1/4 * (2*1/2 - 1/4) = 3/16 = 0.1875. Am I missing something?
I don't know what you mean when you say that my tests are dependent -- I'm not reusing coin flips. Would it be better if I tested two different coins with the null hypothesis being "both coins are fair"? The numbers remain the same, of course. Charles Greathouse Analyst/Programmer Case Western Reserve University
--oh. Sorry, I misunderstood you (Ch.Gr.); stupidly thought you were re-using flips.. Sorry, the real explanation is, your N tests (N=2 for you) are not interchangeable. My problem implicitly assumed they were.