Chapter 3.3, Problem 45BB

Why Divide by n − 1? Let a population consist of the values 2 min, 3 min, 8 min. (These are departure delay times taken from American Airlines flights from New York’s JFK airport to Los Angeles. See Data Set 15 in Appendix B.) Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.)

1. a. Find the variance σ² of the population {2 min, 3 min, 8 min}.
2. b. After listing the nine different possible samples of two values selected with replacement, find the sample variance s² (which includes division by n − 1) for each of them; then find the mean of the nine sample variances s².
3. c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by n), then find the mean of those nine population variances.
4. d. Which approach results in values that are better estimates of σ²: part (b) or part (c)? Why? When computing variances of samples, should you use division by n or n − 1?
5. e. The preceding parts show that s² is an unbiased estimator of σ². Is s an unbiased estimator of σ? Explain.