A service station has six gas pumps. When no vehicles are at the station, let pi denote the probability that the next vehicle will select pump i (i = 1,2,...,6). Based on a sample of size n, we wish to test H0: p1=...=p6 verses Ha: p1 = p3 = p5, p2 = p4 = p6 (note that Ha is not a simple hypothesis). Let X be the number of customers in the sample that select an even-numbered pump.
Show that the likelihood ratio test rejects H0 if either X ≥ c or X ≤ n -c. (Hint: When Ha is true, let θ denote the common value of p2, p4, and p6.)