Chapter 7, Problem 57SE

In Example 6.8, we introduced the concept of a censored experiment in which n components are put on test and the experiment terminates as soon as r of the components have failed. Suppose component lifetimes are independent, each having an exponential distribution with parameter λ. Let Y₁, denote the time at which the first failure occurs, Y₂ the time at which the second failure occurs, and so on, so that T_r = Y₁ + ⋯ + Y_r + (n − r)Y_r is the total accumulated lifetime at termination. Then it can be shown that 2λT_r has a chi-squared distribution with 2r df. Use this fact to develop a 100(1 − α)% CI formula for true average lifetime 1/λ. Compute a 95% CI from the data in Example 6.8.