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## 9Th_Ed Test Bank Statistics for the Behavioral Sciences – Gravetter

9Th_Ed Test Bank Statistics for the Behavioral Sciences – Gravetter

**Sample**

Chapter 7: The Distribution of Sample Means

**Chapter Outline**

7.1 Samples and Populations

7.2 The Distribution of Sample Means

The Central Limit Theorem

The Shape of the Distribution of Sample Means

The Mean of the Distribution of Sample Means: The Expected Value of M

The Standard Error of M

Three Different Distributions

7.3 Probability and the Distribution of Sample Means

A z-Score for Sample Means

7.4 More about Standard Error

In the Literature – Reporting Standard Error

7.5 Looking Ahead to Inferential Statistics

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**Learning Objectives and Chapter Summary**

- Students should be able to define the distribution of sample means and, for a specific sampling situation, describe the distribution by identifying its shape, the expected value of M, and the standard error of M.

The distribution of sample means is the set of sample means for all the possible random samples of a specific size from a specific population. The distribution of sample means is normal if (a) the population from which the samples are obtained is normal, or (b) the sample size is at least n = 30. If samples of size n are selected from a population with a mean of μ and a standard deviation of σ, then the mean of the distribution of sample means (called the expected value of M) is equal to μ and the standard deviation of the distribution of sample mean (called the standard error of M) is equal to σ/Ön.

Note that the standard error of M measures the standard distance between a sample mean, M, and the population mean μ. Thus, the standard error provides a measure of how accurately, on average, a sample mean represents the population mean.

- Students should understand that each sample mean, M, has a location in the distribution of sample means that can be described by a z-score equal to

M – μ

z = ───── where σ_{M} is the standard error of M

σ_{M}

- Using the distribution of sample means, z-scores, and the unit normal table, students should be able to determine probabilities corresponding to specific sample means.

The definition of a z-score as a location within a distribution is the same for sample means as it was for scores (X values) in Chapter 5. Also, the unit normal table is used to find probabilities for sample means the same as it was for scores in Chapter 6. The new element that is introduced in Chapter 7 is that we are now finding z-scores and probabilities for the distribution of sample means.

**Other Lecture Suggestions**

- In Chapter 1, we used Figure 1.2 to illustrate the concept of sampling error. The same figure can be used to help introduce the distribution of sample means and the concept of standard error. First, the figure shows that different samples will have different means, so a large set of samples will produce a large set of sample means (an entire distribution). Second, the figure shows that a sample mean will typically be different from its population mean: M μ. The standard error provides a measure of how much difference (on average) there will be between M and μ.

- Figure 7.7 also provides a concrete illustration of the distribution of sample means. In the figure, each box represents a different sample. Thus, the figures shows that (a) There are hundreds (or thousands) of different samples, and (b) Some of the sample means are very close to μ, but others are far away.

- On a theoretical level, the material presented in Chapter 7 is a major advance over the basic information that was contained in Chapters 5 and 6. On a practical level, however, Chapter 7 introduces only one small new element. Specifically, students must learn that when they are computing z-scores and finding probabilities for sample means, they must use the standard error instead of the standard deviation.

- Although it may seem like an obvious point, remind students that the size of the sample matters. A sample of 100 people is going to be more accurate (more representative) than a sample of 10 people. Therefore, whenever students are working on a problem that involves a sample, the size of the sample, n, has to appear in the calculations. Specifically, the sample size, n, is included in the calculation of the standard error. Figure 7.3 shows the relationship between sample size and standard error: As sample size gets bigger and bigger, the standard error gets smaller and smaller.

# Exam Items for Chapter 7

## Multiple-Choice Questions

- What term is used to identify the mean of the distribution of sample means?
- the expected value of M
- the standard error of M
- the sample mean
- the central limit mean

- What term is used to identify the standard deviation of the distribution of sample means?
- the expected value of M
- the standard error of M
- the sample mean
- the central limit mean

- (www) For a population with µ = 80 and σ = 20, the distribution of sample means based on n = 16 will have an expected value of ____ and a standard error of ____.
- 5, 80
- 80, 5
- 20, 20
- 80, 1.25

- The distribution of sample means ____ .
- is always a normal distribution.
- will be normal
*only if*the population distribution is normal. - will be normal
*only if*the sample size is at least n = 30. - we be normal
*if either*the population is normal or the sample size is n__>__30.

- A sample of n = 100 scores is selected from a population with μ = 80 with σ = 20. On average, how much error is expected between the sample mean and the population mean?
- 0.2 points
- 0.8 points
- 2 points
- 4 points

- A sample of n = 16 scores is selected from a population with μ = 80 with σ = 20. On average, how much error would be expected between the sample mean and the population mean?
- 20 points
- 5 points
- 4 points
- 1.25 points

- What symbol is used to identify the standard error of M?
- σ
_{M} - µ
- σ
- M
_{M}

- (www) Under what circumstances will the distribution of sample means be normal?
- It is always normal.
- only if the population distribution is normal
- if the sample size is greater than 30
- if the population is normal or if the sample size is greater than 30.

- A random sample of n = 4 scores is selected from a population. Which of the following distributions definitely will be normal?
- The scores in the sample will form a normal distribution.
- The scores in the population will form a normal distribution
- The distribution of sample means will form a normal distribution.
- Neither the sample, the population, nor the distribution of sample means will

definitely be normal.

- (www) A random sample of n = 36 scores is selected from a population. Which of the following distributions definitely will be normal?
- The scores in the sample will form a normal distribution.
- The scores in the population will form a normal distribution
- The distribution of sample means will form a normal distribution.
- Neither the sample, the population, nor the distribution of sample means will

definitely be normal.

- Samples of size n = 9 are selected from a population with μ = 80 with σ = 18. What is the expected value for the distribution of sample means?
- 6
- 18
- 80/3
- 80

- Samples of size n = 9 are selected from a population with μ = 80 with σ = 18. What is the standard error for the distribution of sample means?
- 80
- 18
- 6
- 2

- (www) If random samples, each with n = 9 scores, are selected from a normal population with µ = 80 and σ = 36, then what is the expected value for the distribution of sample means?
- 4
- 12
- 16
- 80

- If random samples, each with n = 4 scores, are selected from a normal population with µ = 80 and σ = 36, then what is the standard error for the distribution of sample means?
- 4
- 9
- 18
- 36

- If all the possible random samples with n = 36 scores are selected from a normal population with µ = 80 and σ = 18, and the mean is calculated for each sample, then what is the average of all the sample means?
- 2
- 6
- 80
- cannot be determined without additional information

- If random samples, each with n = 36 scores, are selected from a normal population with µ = 80 and σ = 18, how much difference, on average, should there be between a sample mean and the population mean?
- 2 points
- 3 points
- 6 points
- 18 points

- What happens to the expected value of M as sample size increases?
- It also increases.
- It decreases.
- It stays constant.
- The expected value does not change in a predictable manner when sample size

increases.

- What happens to the standard error of M as sample size increases?
- It also increases.
- It decreases.
- It stays constant.
- The standard error does not change in a predictable manner when sample size

increases.

- Which combination of factors will produce the smallest value for the standard error?
- a large sample and a large standard deviation
- a small sample and a large standard deviation
- a large sample and a small standard deviation
- a small sample and a small standard deviation

- (www) For a particular population, a sample of n = 9 scores has a standard error of 8. For the same population, a sample of n = 16 scores would have a standard error of _____.
- 8
- 6
- 4
- 2

- For a particular population a sample of n = 4 scores has an expected value of 10. For the same population, a sample of n = 25 scores would have an expected value of _____.
- 4
- 8
- 10
- 20

- (www) A sample of n = 9 scores has a standard error of 6. What is the standard deviation of the population from which the sample was obtained?
- 54
- 18
- 6
- 2

- A sample obtained from a population with σ = 12 has a standard error of 2 points. How many scores are in the sample?
- n = 36
- n = 24
- n = 6
- n = 3

- A sample of n = 4 scores has a standard error of 10 points. For the same population, what is the standard error for a sample of n = 16 scores?
- 1
- 2.5
- 5
- 10

- A random sample of n = 4 scores is obtained from a population with a mean of µ = 80 and a standard deviation of σ = 10. If the sample mean is M = 90, what is the z-score for the sample mean?
- z = 20.00
- z = 5.00
- z = 2.00
- z = 1.00

- A sample of n = 4 scores is selected from a population with μ = 50 and σ = 12. If the sample mean is M = 56, what is the z-score for this sample mean?
- 0.50
- 1.00
- 2.00
- 4.00

- For a normal population with a mean of µ = 80 and a standard deviation of σ = 10, what is the probability of obtaining a sample mean greater than M = 75 for a sample of n = 25 scores?
- p = 0.0062
- p = 0.9938
- p = 0.3085
- p = 0.6915

- A sample of n = 16 scores is obtained from a population with μ = 50 and σ = 16. If the sample mean is M = 54, then what is the z-score for the sample mean?
- z = 0.25
- z = 0.50
- z = 1.00
- z = 4.00

- (www) A sample of n = 9 scores is obtained from a population with μ = 70 and σ = 18. If the sample mean is M = 76, then what is the z-score for the sample mean?
- z = 0.33
- z = 0.50
- z = 1.00
- z = 3.00

- If a sample of n = 4 scores is obtained from a population with μ = 70 and σ = 12, then what is the z-score corresponding to a sample mean of M = 76?
- z = 0.25
- z = 0.50
- z = 1.00
- z = 2.00

- A sample of n = 4 scores is obtained from a population with μ = 70 and σ = 8. If the sample mean corresponds to a z-score of 2.00, then what is the value of the sample mean?
- M = 86
- M = 78
- M = 74
- M = 72

- A sample from a population with μ = 40 and σ = 10 has a mean of M = 44. If the sample mean corresponds to a z = 2.00, then how many scores are in the sample?
- n = 100
- n = 25
- n = 5
- n = 4

- A random sample of n = 16 scores is obtained from a population with σ = 12. If the sample mean is 6 points greater than the population mean, what is the z-score for the sample mean?
- +6.00
- +2.00
- +1.00
- cannot be determined without knowing the population mean

- For a normal population with µ = 40 and σ = 10 which of the following samples is
*least likely*to be obtained? - M ≤ 42 for a sample of n = 4
- M ≤ 44 for a sample of n = 4
- M ≤ 42 for a sample of n = 100
- M ≤ 44 for a sample of n = 100

- For a normal population with µ = 40 and σ = 10 which of the following samples has the highest probability of being obtained?
- M ≤ 42 for a sample of n = 4
- M ≤ 44 for a sample of n = 4
- M ≤ 42 for a sample of n = 100
- M ≤ 44 for a sample of n = 100

- (www) A random sample of n = 4 scores is obtained from a normal population with µ = 20 and σ = 4. What is the probability that the sample mean will be greater than M = 22?
- 0.50
- 1.00
- 0.1587
- 0.3085

- (www) A random sample of n = 9 scores is obtained from a normal population with µ = 40 and σ = 6. What is the probability that the sample mean will be greater than M = 43?
- 0.3085
- 0.6915
- 0.9332
- 0.0668

- A random sample of n = 16 scores is obtained from a normal population with µ = 40 and σ = 8. What is the probability that the sample mean will be within 2 points of the population mean?
- 0.3830
- 0.6826
- 0.8664
- 0.9544

- A sample is obtained from a population with μ = 100 and σ = 20. Which of the following samples would produce the most extreme z-score?
- a sample of n = 25 scores with M = 102
- a sample of n = 100 scores with M = 102
- a sample of n = 25 scores with M = 104
- a sample of n = 100 scores with M = 104

- (www) A sample of n = 16 scores is selected from a population with μ = 100 and σ = 32. If the sample mean is M = 104, what is the z-score for this sample mean?
- 2.00
- 1.00
- 0.50
- 0.25

## True/False Questions

- The mean for the distribution of sample means is always equal to the mean for the population from which the samples are obtained.

- In order for the distribution of sample means to be normal, it must be based on samples of at least n = 30 scores.

43 A sample of n = 25 scores is selected from a population with a mean of µ = 80 and a standard deviation of σ = 20. The expected value for the sample mean is 80.

- A sample of n = 25 scores is selected from a population with a mean of µ = 80 and a standard deviation of σ = 20. The standard error for the sample mean is 20.

- Two samples probably will have different means even if they are both the same size and they are both selected from the same population.

- As the sample size increase, the standard error also increases.

- The mean for a sample of n = 4 scores has a standard error σ = 5 points. This sample was selected from a population with a standard deviation of σ = 20.

- If samples are selected from a normal population, the distribution of sample means will also be normal.

- According to the central limit theorem, the standard error for a sample mean becomes smaller as the sample size increases.

- If samples of size n = 16 are selected from a population with μ = 40 and σ = 8, the distribution of sample means will have an expected value of 40.

- If samples of size n = 16 are selected from a population with μ = 40 and σ = 8, the distribution of sample means will have a standard error of 2 points.

- If the sample size is equal to the standard deviation (n = s), then the standard error is equal to the square root of n.

- The mean for a sample of n = 9 scores has a standard error of 2 points. This sample was selected from a population with a standard deviation of σ = 18.

- As the population standard deviation increases, the standard error will also increase.

- If the standard deviation for a population increases, the standard error for sample means from the population will also increase.

- The smallest possible standard error is obtained when a small sample is taken from a population with a small standard deviation.

- On average, a sample of n = 16 scores from a population with σ = 10 will provide a better estimate of the population mean than you would get with a sample of n = 16 scores from a population with σ = 5.

- A sample is obtained from a population with σ = 20. If the sample mean has a standard error of 5 points, then the sample size is n = 4.

- If the sample size is equal to the population standard deviation (n = s), then the standard error for the sample mean is equal to 1.00.

- A researcher obtained M = 27 for a sample of n = 36 scores selected from a population with µ = 30 and σ = 18. This sample mean corresponds to a z‑score of z = –1.00.

- A sample of n = 9 scores is selected from a normal population with a mean of µ = 80 and a standard deviation of σ = 12. The probability that the sample mean will be greater than M = 86 is equal to the probability of obtaining a z-score greater than z = 1.50

- A sample of n = 9 scores is randomly selected from a population with µ = 80 and

σ = 9. If the sample mean is M = 83, then the corresponding z‑score is z = +3.00.

- A sample of n = 4 scores is selected from a population with μ = 30 and σ = 8. The probability of obtaining a sample mean greater than 34 is equal to the probability of obtaining a z-score greater than z = 2.00 from a normal distribution.

- A sample of n = 4 scores is selected from a population with μ = 70 and σ = 10. The probability of obtaining a sample mean greater than 65 is p = 0.8413.

- A sample of n = 9 scores is selected from a population with μ = 50 and σ = 12. The probability of obtaining a sample mean greater than 46 is p = 0.8413.

- A sample of n = 25 scores is selected from a population with μ = 50 and σ = 10. The probability of obtaining a sample mean greater than 55 is p = 0.3085.

- A sample of n = 16 scores is selected from a population with μ = 70 and σ = 8. It is very unlikely that the sample mean will be greater than 78.

- A sample of n = 25 scores is selected from a population with μ = 70 and σ = 20. It is very unlikely that the sample mean will be smaller than 72.

- A population has µ = 60 and σ = 30. For a sample of n = 25 scores from this population, a sample mean of M = 55 would be considered an extreme value.

- A population has µ = 60 and σ = 10. For a sample of n = 25 scores from this population, a sample mean of M = 55 would be considered an extreme value.

## Other Exam Items

- Define the
*distribution of sample means*.

- Describe what is measured by the standard error of M.

- Describe the shape, the mean, and the standard deviation for each of the following two distributions.
- A population of scores with µ = 50 and σ = 6.
- The distribution of sample means based on samples of n = 36 selected from a

population with µ = 50 and σ = 6.

- A population has a mean of µ = 80 with σ = 20.
- If a single score is randomly selected from this population, how much distance, on

average, should you find between the score and the population mean?

- If a sample of n = 4 scores is randomly selected from this population, how much

distance, on average, should you find between the sample mean and the population

mean?

- If a sample of n = 100 scores is randomly selected from this population, how much

distance, on average, should you find between the sample mean and the population

mean?

- Each of the following samples was obtained from a population with µ = 100 and σ = 10. Find the z-score corresponding to each sample mean.
- M = 95 for a sample of n = 4
- M = 104 for a sample of n = 25
- M = 103 for a sample of n = 100

- A sample of n = 16 scores is selected from a normal population with µ = 40 and σ = 12.
- Describe the distribution of sample means that contains the mean for the sample.
- What is the probability that the sample mean will be greater than 43?
- What is the probability that the sample mean will be less than 34?

- (www) For a normal population with µ = 100 and σ = 20,
- What is the probability of obtaining a sample mean greater than 110 for a sample of

n = 4 scores?

- What is the probability of obtaining a sample mean greater than 110 for a sample of n = 16 scores?
- For a sample of n = 25 scores, what is the probability that the sample mean will be

within 5 points of the population mean? In other words, what is p(95 < M < 105)?

## Answers for Multiple-Choice Questions (with section and page numbers from the text)

- a, 7.2, p. 206 11. d, 7.2, p. 206 21. c, 7.2, p. 206 31. b, 7.3, p. 213
- b, 7.2, p. 207 12. c, 7.2, p. 207 22. b, 7.2, p. 207 32. b, 7.3, p. 213
- b, 7.2, p. 207 13. d, 7.2, p. 206 23. a, 7.2, p. 207 33. b, 7.3, p. 213
- d, 7.2, p. 205 14. c, 7.2, p. 207 24. c, 7.2, p. 207 34. a, 7.3, p. 213
- c, 7.2, p. 207 15. c, 7.2, p. 206 25. c, 7.3, p. 213 35. d, 7.3, p. 213
- b, 7.2, p. 207 16. b, 7.2, p. 207 26. b, 7.3, p. 213 36. c, 7.3, p. 211
- a, 7.2, p. 207 17. c, 7.2, p. 206 27. b, 7.3, p. 211 37. d, 7.3, p. 211
- d, 7.2, p. 205 18. b, 7.2, p. 207 28. c, 7.3, p. 213 38. b, 7.3, p. 211
- d, 7.2, p. 209 19. c, 7.2, p. 207 29. c, 7.3, p. 213 39. d, 7.3, p. 213
- c, 7.2, p. 209 20. b, 7.2, p. 207 30. c, 7.3, p. 213 40. c, 7.3, p. 213

## Answers for True/False Questions (with section and page numbers from the text)

- T, 7.2, p. 206 51. T, 7.2, p. 207 61. T, 7.3, p. 213
- F, 7.2, p. 205 52. T, 7.2, p. 207 62. F, 7.3, p. 213
- T, 7.2, p. 206 53. F, 7.2, p. 207 63. F, 7.3, p. 213
- F, 7.2, p. 207 54. T, 7.2, p. 207 64. T, 7.3, p. 211
- T, 7.1, p. 201 55. T, 7.2, p. 207 65. T, 7.3, p. 211
- F, 7.2, p. 207 56. F, 7.2, p. 207 66. F, 7.3, p. 211
- F, 7.2, p. 207 57. F, 7.2, p. 207 67. T, 7.3, p. 213
- T, 7.2, p. 205 58. F, 7.2, p. 207 68. F, 7.3, p. 213
- T, 7.2, p. 205 59. F, 7.2, p. 207 69. F, 7.3, p. 213
- T, 7.2, p. 206 60. T, 7.3, p. 213 70. T, 7.3, p. 213

## Answers for Other Exam Items

- The distribution of sample means is the set of sample means obtained from all the possible random samples of a specified size (n) taken from a particular population.

- The standard error of M is the standard deviation for the distribution of sample means, and measures the standard distance (or deviation) between a sample mean M and the population mean µ.

- a. The population is a distribution of scores with an unknown shape, a mean of µ = 50 and a

standard deviation of σ = 6.

- The distribution of sample means is normal (because n > 30), with a mean (expected value) of µ = 50, and a standard deviation (standard error) of σ
_{M}= 1.

- a. σ = 20 points
- σ
_{M}= 10 points - σ
_{M}= 2 points

- a. z = –1.00
- z = +2.00
- z = +3.00

- a. The distribution will be normal (because the sample size is greater than 30), will have an

expected value of µ = 40 and a standard error of 12/4 = 3 points.

- z = 1.00, p = 0.1587
- z = –2.00, p = 0.0228

- a. z = 10/10 = 1.00. p = 0.1587
- z = 10/5 = 2.00. p = 0.0228
- z = ±5/4 = ±1.25. p = 2(0.3944) = 0.7888

9Th_Ed Test Bank Statistics for the Behavioral Sciences – Gravetter

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