In Example 6.8, we introduced the concept of a censoredexperiment in which n components

Chapter 7, Problem 57

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In Example 6.8, we introduced the concept of a censored experiment in which n components are put on test and experiment terminates as soon as r of the components have failed. Suppose component lifetimes are independent, each having an exponential distribution with parameter \(\gamma\). 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 \(T_{r}=Y_{1}+\cdots+Y_{r}+(n-r) Y_{r}\) is the total accumulated lifetime at termination.Then it can be shown that 2lTr has a chi-square distribution with 2r df. Use this fact to develop a100(1 - \(\alpha\))% CI formula for true average lifetime 1/\(\lambda\).Compute a 95% CI from the data in Example 6.8.