Chapter 8, Problem 79SE

Let X₁, X₂,… X_n be a random sample from an exponential distribution with parameter λ. Then it can be shown that 2λΣX_i has a chi-squared distribution with v = 2n (by first showing that 2λX_i has a chi-squared distribution with v = 2).

1. a. Use this fact to obtain a test statistic for testing H₀: μ = μ₀. Then explain how you would determine the P-value when the alternative hypothesis is H_a: μ < μ₀. [Hint: E(X_i) = μ = 1/λ. So μ = μ₀. equivalent to λ = 1/μ₀.]
2. b. Suppose that ten identical components, each having exponentially distributed time until failure, are tested. The resulting failure times are

   

3. c. 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.]