Wayne Collier designed an experiment to measure the fuel efficiency of his family cat under different tire pressures. For each run, he set the tire pressure and then measured the miles he drove on a highway (I-95 between Mills River and Pisgah Forest, NC) until he ran out of fuel using 2 liters of fuel each time. To do this, he made some alterations to the normal flow of gasoline to the engine.
In Wayne's words, "I inserted a T-junction into the fuel line just before the fuel filter, and a line into the passenger compartment of my car, where it joined with a graduated 2 liter Rubbermaid bottle that I mounted in a box where the passenger seat is normally fastened. Then I sealed off the fuel-return line, which under normal operation sends excess fuel from the fuel pump back to the fuel tank."
[Problem] Suppose that you call the mean miles that he can drive with normal pressure in the tires MU (mean). An unbiased estimate for MU is the mean of the sample runs, x-bar. But Wayne has a different idea. He decides to use the following estimator: He flips a fair coin. If the coin comes up heads, he will add five miles to each observation. If tails come up, he will subtract five miles from each observation.
(a) Show that Wayne's estimate is, in fact, unbiased.
(b) Compare the standard deviation of Wayne's estimate with the standard deviation of the sample mean.
(c) Given your answer to (b), why does Wayne's estimate not make good sense scientifically?