Three Randomly Selected Households Are Surveyed as a Pilot Project

Essay

Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 5, 6, and 10. Consider the values of 5, 6, and 10 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 5, 6, and 10. The nine different samples are as follows:
(5, 5), (5, 6), (5, 10), (6, 5), (6, 6), (6, 10), (10, 5), (10, 6), and (10, 10).
(i) Find the variance of each of the nine samples, then summarize the sampling distribution of the variances in the format of a table representing the probability distribution.
(ii) Compare the population variance to the mean of the sample variances.
(iii) Do the sample variances target the value of the population variance? In general, do variances make good estimators of population variances? Why or why not?