Three Randomly Selected Households Are Surveyed as a Pilot Project

question 71

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 1, 3, and 9. Consider the values of 1, 3, and 9 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 1, 3, and 9. The nine different samples are as follows: (1, 1), (1, 3), (1, 9), (3, 1), (3, 3), (3, 9), (9, 1), (9, 3), and (9, 9).
(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?

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Definitions:

Probability

A quantification of an event's probability, on a scale from 0 to 1, where a 0 denotes the event is impossible and a 1 denotes the event is assured to occur.

Foreclosed

A legal process where a lender takes control of a property from a borrower who has failed to meet the terms of their loan.

Vickrey Auction

A sealed-bid auction where the highest bidder wins but pays the amount bid by the second-highest bidder.

Willingness To Pay

The maximum amount an individual is ready to spend to acquire a good or service.