A sample of n captured Pandemonium jet fighters results in serial numbers x1, x2, x3, . . . , xn. The CIA knows that the aircraft were numbered consecutively at the factory starting with α and ending with β, so that the total number of planes manufactured is β ❝ α + 1 (e. g., if α = 17 and β = 29, then 29 ❝ 17 + 1 = 13 planes having serial numbers 17, 18, 19, . . . , 28, 29 were manufactured.) However, the CIA does not know the values of α or β. A CIA statistician suggests using the estimator max(Xi) ❝ min(Xi) + 1 to estimate the total number of planes manufactured. Required:
a. If n = 5, x1 = 201, x2 = 350, x3 = 415, x4 = 472, and x5 = 414, what is the corresponding estimate?
b. Under what conditions on the sample will the value of the estimate be exactly equal to the true total number of planes?