An American roulette wheel is spun n = 3880 times in order to test if it is fair (i. e. to test if each slot has probability 1/38). Suppose that each of the 36 numbered slots (1, 2,.., 36) comes up exactly 100 times and each of "0" and "00" comes up 140 times.

(a) Test at the 5% level using the X2 test if the wheel is fair.
(b) Now suppose that before you had looked at the data you had suspected that the numbered slots were less likely than the "0" and "00", and you had decided to test the binomial hypothesis H0: P("0" or "00") = 2/38 versus H1: P("0" or "00") > 2/38. We know that the UMP test of these hypotheses rejects H0 if Z (= total number of "0" and "00"s) is greater than C, where C is chosen for a level 0.05 test. Use the fact that under H0 Z has approximately a Normal N(n*(2/38), n*(2/38)*(36/38)) distribution [this follows from the Central Limit Theorem, for example] to find C and perform the test.
(c) Compare the result in (b) with that in (a). (This is intended to illustrate that with exactly the same data different conclusions may be reached depending upon how focused the test is upon a narrow hypothesis. The test in (a) tests against all alternatives to a fair wheel and the test of (b) is focused narrowly upon deviations of the probabilities of "0" and "00" slots from the others. As this example should suggest, it would be statistically impermissible to focus the test after seeing the data without compensating for this choice of a narrow hypothesis in some way.)