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 the 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 s? Explain.