Question

Use the population of ages {56, 49, 58, 46} of the four U.S. presidents (Lincoln, Garfield, McKinley, Kennedy) when they were assassinated in office. Assume that random samples of size n = 2 are selected with replacement.

1. List the 16 different samples. For example, the samples for age 56 would be

56, 56

56, 49

56, 58

56, 46.

2. After listing all 16 samples, find the mean of each sample, then construct a table representing the sampling distribution of the sample mean. In the table, combine values of the sample mean that are the same.

3. Compare the mean of the population {56, 49, 58, 46} to the mean of the sampling distribution of the sample mean.

4. Do the sample means target the value of the population mean? In general, do sample means make good estimators of population means? Why or why not?

Answer 1

1. 16 different samples are:

(56, 56), (56, 49), (56, 58), (56, 46), (49, 56), (49, 49), (49, 58), (49, 46), (58, 56), (58, 49), (58, 58), (58, 46), (46, 56), (46, 49),

(46, 58), (46, 46)

2 Samples Sample mean(= ) (56, 56) (56, 49) (56, 58) (56, 46) (49, 56) (49, 49) (49, 58) (49, 46) (58, 56) (58, 49) (58, 58) (58, 46) (46, 56) (46, 49) (46, 58) (46, 46) 56 52.5 57 51 52.5 49 53.5 47.5 57 53.5 58 52 51 47.5 52 46 Sampling distribution of X sample mean 52.5 53.5 Total 46 47.5 49 51 52 56 57 58 )1/16 2/16 P(X 3. E(X) 46* (1/16)+47.5* (2/16)+ 49* (1/16)+51 (2/16)+52* (2/16) 52.5 (2/16)53.5 (2/16)56* (1/16)+57* (2/16)58 (1/16) 52.25 Population Hence population mean and mean of sampling distribution of sample 1/16 2/16 2/16 2/16 2/16 1/16 2/16 1/16 1 (56 49 58 +46)/4 52.25 mean mean are same

4. Yes. In general, sample means are good estimators of population means. Since sample mean is best linear unbiased estimator (BLUE) of population mean.