91. Let X1, X2, . . . , Xn be a random sample from an exponential distribution with parameter l. Then it can be shown that has a chi-squared distribution with n  2n (by rst showing that 2lXi has a chisquared distribution with n  2).

a. Use this fact to obtain a test statistic and rejection region that together specify a level a test for H0:

m  m0 versus each of the three commonly encountered alternatives. [Hint: E(Xi)  m  1/l, so m  m0 is equivalent to l  1/m0.]

b. Suppose that ten identical components, each having exponentially distributed time until failure, are tested. The resulting failure times are 95 16 11 3 42 71 225 64 87 123 Use the test procedure of part

(a) to decide whether the data strongly suggests that the true 2lgXi x  6.33 sˆ

u ˆ

u ˆ

su ˆ V1u ˆ 2 X

X E1X2  m V1X2  s2/n.

average lifetime is less than the previously claimed value of 75.