83. Let X1, X2, . . . , Xn be a random sample from an exponential distribution with parameter . Then it can be shown that 2
Xi has a chi-squared distribution with
2n (by first showing that 2Xi has a chi-squared distribution with
2).

a. Use this fact to obtain a test statistic and rejection region that together specify a level  test for H0:
0 versus each of the three commonly encountered alternatives.

[Hint: E(Xi

)

1/, so
0 is equivalent to


1/0.]

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