Solved: Let X1, X2, …., Xn be a random sample from an

Chapter 8, Problem 79E

(choose chapter or problem)

Let \(X_1, X_2, \ldots, X_n\) be a random sample from an exponential distribution with parameter \(\lambda\). Then it can be shown that \(2 \lambda \Sigma X_i\) has a chi-squared distribution with \(\nu=2 n\) (by first showing that \(2 \lambda X_i\) has a chi-squared distribution with v=2).

a. Use this fact to obtain a test statistic for testing \(H_0: \mu=\mu_0\). Then explain how you would determine the P-value when the alternative hypothesis is \(H_{\mathrm{a}}: \mu<\mu_0\). [Hint: \(E\left(X_i\right)=\mu=1 / \lambda\), so \(\mu=\mu_0\) is equivalent to \(\lambda=1 / \mu_0\).]

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 1  23

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 . [Hint: Consult Table A.7.]