Given the following very simple population distribution, construct the theo¬

retical sampling distribution for the sample mean based on three independent observations from the population.

x P(X = x)

1 1/3 2 1/3 3 1/3

[Hint: Enumerate the possible combinations of events for samples of size 3, and then translate these into values of the sample mean.J Similarly, find the distribution of the sample median and the sample mid-range (one-half times the sum of the largest value in the sample and the smallest value in the sample). Using these sampling distributions,

(a) Show that the sample mean, the sample median, and the sample mid¬

range are all unbiased estimators of the population mean.

(b) Find the variance of the sample mean, the sample median, and the sample mid-range.

(c) Find the relative efficiency of the sample mean as compared with the sample median and the relative efficiency of the sample mean as com¬

pared with the sample mid-range.