Question

Wayne Collier designed an experiment to measure the
fuel efficiency of his family car 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
d©
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.”
Suppose that you call the mean miles that he can drive with
µ.
µ
normal pressure in the tires
An unbiased estimate for
is the
mean of the sample runs, x. 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

Answer 1