Question: A researcher considered a single-machine-tool system with repair times (in minutes) that can be modeled by an exponential distribution with \(\theta \) = 60. Of interest is the mean repair time, \(\bar{X}\) of a sample of 100 machine breakdowns.

Find the mean of \(\bar{X}\) and the variance of \(\bar{X}\).

What probability distribution provides the best model of the sampling distribution of \(\bar{X}\) ? Why?

Calculate the probability that the mean repair time, \(\bar{X}\) is no longer than 30 minutes.