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Elementary Statistics | 12th Edition | ISBN: 9780321836960 | Authors: Mario F. Triola
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In Exercises, use the population of ages {56, 49, 58, 46}

Chapter 6, Problem 11BSC

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QUESTION:

In Exercises 11–14, 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.

Sampling Distribution of the Sample Mean

a. After identifying the 16 different possible 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. (Hint: See Table 6­4 in Example 1.)

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

c. 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?

Questions & Answers

QUESTION:

In Exercises 11–14, 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.

Sampling Distribution of the Sample Mean

a. After identifying the 16 different possible 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. (Hint: See Table 6­4 in Example 1.)

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

c. 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:

Answer:

Step 1 of 2

a. The sample mean is the numbers in the sample divided by number of values in the sample

n = 2.

Add the probabilities for each sample value.

Thus the second table is the sampling distribution

Sample

Sample Mean

Probability

56, 56

56

1/16

56, 49

52.5

1/16

56, 58

57

1/16

56, 46

51

1/16

49, 56

52.5

1/16

49, 49

49

1/16

49, 58

53.5

1/16

49, 46

47.5

1/16

58, 56

57

1/16

58, 49

53.5

1/16

58, 58

58

1/16

58, 46

52

1/16

46, 56

51

1/16

46, 49

47.5

1/16

46, 58

52

1/16

46, 46

46

1/16

Sample Mean

Probability

46

1/16 = 0.0625

47.5

2/16 = 0.125

49

1/16 = 0.0625

51

2/16 = 0.125

52

2/16 = 0.125

52.5

2/16 = 0.125

53.5

2/16 = 0.125

56

1/16 = 0.0625

57

2/16 = 0.125

58

0.0625

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