A sample of $n$ captured jet fighters results in serial numbers $x_{1}, x_{2}, x_{3}, \ldots, x_{n} .$ The CIA knows that the aircraft were numbered consecutively at the factory starting with $\alpha$ and ending with $\beta$ , so that the total number of planes manufactured is $\beta-\alpha+1$ (e.g., if $\alpha=17$ and $\beta=29$ , then $29-17+1=13$ planes having serial numbers $17,18,19, \ldots, 28,29$ were manufactured $)$ However, the CIA does not know the values of $\alpha$ or $\beta .$ A CIA statistician suggests using the estimator max $\left(X_{i}\right)-\min \left(X_{i}\right)+1$ to estimate the total number of planes manufactured.

(a) If $n=5, x_{1}=237, \quad x_{2}=375, \quad x_{3}=202, \quad x_{4}=525,$ and $x_{5}=418,$ 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 $\beta-\alpha+1 ?$ Explain in one or two sentences.

(A similar method was used to estimate German tank production in World War II.)