Consider the series of examples in this section concerning service times in a queue. Suppose that the manager observes two service times X1 and X2. It is easy to see that both f1(x) in (9.2.1) and f0(x) in (9.2.2) depend on the observed data only through the value t = X1 + X2 of the statistic T = X1 + X2. Hence, the tests from Theorems 9.2.1 and 9.2.2 both depend only on the value of T.
a. Using Theorem 9.2.1, determine the test procedure that minimizes the sum of the probabilities of type I and type II errors.
b. Suppose that X1 = 4 and X2 = 3 are observed. Perform the test in part (a) to see whetherH0 is rejected.
c. Prove that the distribution of T, given thatH0 is true, is the gamma distribution with parameters 2 and 1/2.
d. Using Theorem 9.2.2, determine the test procedure with level at most 0.01 that has minimum probability of type II error. Hint: It looks like you need to solve a system of nonlinear equations, but for a level 0.01 test, the equations collapse to a single simple equation.
e. Suppose that X1 = 4 and X2 = 3 are observed. Perform the test in part (d) to see whetherH0 is rejected.