The waiting time (in days) until the occurrence of a recordable safety incident in a certain company's manufacturing site is known to be an exponential random variable with unknown parameter β. As part of a safety performance characterization program, the following data set was obtained S1 = {16, 1, 9, 34, 63, 44, 1, 63, 41, 29}

(i) Considering this as a speci?c realization of a random sample of size n = 10, determine a numerical value, x¯, for the random sample mean, X¯ , and obtain an exact sampling distribution for the sampling mean in terms of the unknown population parameter, β, and sample size, n.

(ii) The company claims that its true mean waiting time between safety incidents is 40 days. From the sampling distribution and the speci?c value x¯ obtained in (i), determine P (X¯ ≤ x¯|β = 40).

(iii) An independent safety auditor estimates that from the company's records, oper- ating procedures, and other performance characteristics, the true mean waiting time between safety incidents is more likely to be 30 days. Determine P (X¯ ≤ x¯|β = 30).

Compare this probability to the one obtained in (ii) and comment on which postulated value for the true mean waiting time is more believable, β = 40 as claimed by the company, or β = 30 as claimed by the independent safety auditor.