A sample of n captured Pandemonium jet fighters results in serial numbers x1, x2, x3, ... , Xn. The CIA knows that the air craft were numbered consecutively at the factory starting with a and ending with b, 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 a or b. A CIA statistician suggests using the estimator max(xi) - min(xi) + 1 to estimate the total number of planes manufactured.
a. If n = 5, X1 = 237, X2 = 375, x3 = 202, x4 = 525, and x5 = 418, 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? Will the estimate ever be larger than the true total? Do you think the estimator is unbiased for estimating β - α + 1? Explain in one or two sentences.