Chapter 3.2, Problem 45BB

Why Divide by n − 1? Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). 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.)

a. Find the variance σ² of the population {9 cigarettes, 10 cigarettes, 20 cigarettes}.

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 then mean of the nine sample variances s².

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.

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?

e. The preceding parts show that s² is an unbiased estimator of σ². Is s an unbiased estimator of σ? Explain.