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

population of ages {56, 49, 58, 46}

1. List the 16 different samples-

{( 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). (49,58), (46,46)}

2.

Sample mean,	P()
46	1/16
47.5	2/16
49	1/16
51	2/16
52	1/16
52.5	2/16
53.5	3/16
56	1/16
57	2/16
58	1/16

3. Population mean , = 209/4 =52.25

Mean of the sampling distribution

= 46 * (1/16) + 47.5 *(2/16) + 49* (1/16) + 51* (2/16) + 52* (1/16) + 52.5 * (2/16) + 53.5 * (3/16) + 56* (1/16)+ 57*(2/16) + 58* (1/16)

= 52.34

Here in this case both the population mean and the sample mean are approxiately same.

4. Yes, the sample mean targets the population mean., i.e, sample means make good estimators of population means. Because here our both sample mean and population mean are same almost.