1. Explain what a sampling distribution is. Why are sampling distributions so important? (p. 396)

2. For a normal population, what can we say about the sampling distribution of the sample mean? (p. 399)

3. True or false: ${\mu }_{\overline{x}}=\mu$ and $\hspace{0.17em}{\sigma }_{\overline{x}}=\sigma /\sqrt{n}$ regardless of whether or not the sampling distribution of $\overline{x}$ is normal. (pp. 397, 398)

4. Use the Central Limit Theorem to explain what happens to the sampling distribution of $\overline{x}$ as the sample size gets larger. (p. 401)

5. According to our rule of thumb, what is the minimum sample size for approximate normality of the sampling distribution of $\overline{x}$? (p. 401)

6. State the three possible cases for the sampling distribution of $\overline{x}$. (p. 401)

7. Suppose we want to decrease the size of the standard error to half its original size. How much do we have to increase the sample size? (p. 398)

8. State the conditions when the sampling distribution of $\overline{x}$ is neither normal nor approximately normal. (p. 401)

1. State the value of the mean of the sampling distribution of $\overline{x}$, that is, ${\mu }_{\overline{x}}=\mu$.
2. Calculate the value of the standard error of the sampling distribution of $\overline{x}$, that is, ${\sigma }_{\overline{x}}=\sigma /\sqrt{n}$.
3. Draw a graph of the distribution curve of $X$, similar to the orange curve in Figure 2 on page 398. On the number line, indicate $\mu ,\hspace{0.17em}\mu -3\sigma$ and $\hspace{0.17em}\mu +3\sigma$.
4. Draw a graph of the distribution curve for the sampling distribution of $\overline{x}$, similar to the green curve in Figure 2 on page 398. On the number line, indicate ${\mu }_{\overline{x}},\hspace{0.17em}{\mu }_{\overline{x}}-3{\sigma }_{\overline{x}}$, and $\hspace{0.17em}{\mu }_{\overline{x}}+3{\sigma }_{\overline{x}}$. Note how both graphs are centered at the population mean, but also note that the variability of the sampling distribution of $\overline{x}$ is less than that of the original distribution of $X$.

9. $\mu =10,\sigma =2,n=25$

10. $\mu =10,\sigma =2,n=100$

11. $\mu =0,\sigma =1,n=9$

12. $\mu =0,\sigma =1,n=25$

13. $\mu =100,\sigma =10,n=100$

14. $\mu =100,\sigma =10,n=400$

15. $\mu =75,\sigma =15,n=36$

16. $\mu =75,\sigma =15,n=25$

17. Systolic blood pressure readings are not normally distributed, with $\mu =78$ and $\hspace{0.17em}\sigma =7$. A random sample of size $n=49$ is taken.

18. Systolic blood pressure readings are not normally distributed, with $\mu =78$ and $\hspace{0.17em}\sigma =7$. A random sample of size $n=25$ is taken.

19. The gas mileage for a 2014 Toyota Prius hybrid vehicle is not normally distributed, with $\mu =55$ miles per gallon and $\sigma =8$. A random sample of size $n=16$ is taken.

20. The gas mileage for a 2014 Toyota Prius hybrid vehicle is not normally distributed, with $\mu =55$ miles per gallon and $\sigma =8$. A random sample of size $n=64$ is taken.

21. The pollen count distribution for Los Angeles in September is not normally distributed, with $\mu =10$ and $\sigma =2$. A random sample of size 16 is taken.

22. The pollen count distribution for Los Angeles in September is not normally distributed, with $\mu =10$ and $\sigma =2$. A random sample of size 64 is taken.

23. Prices for boned trout are normally distributed, with $\mu =\$4.00$ per pound and $\sigma =\$0.50$. A random sample of size 16 is taken.

24. Prices for boned trout are not normally distributed, with $\mu =\$4.00$ per pound and $\sigma =\$0.50$. A random sample of size 36 is taken.

25. Accountant incomes are not normally distributed, with $\mu =\$80,000$ per year and $\sigma =\$20,000$. A random sample of 10 is taken.

26. Accountant incomes are not normally distributed, with $\mu =\$80,000$ per year and $\sigma =\$20,000$. A random sample of 100 is taken.

27. Calculate ${\mu }_{\overline{x}}$ and $\hspace{0.17em}{\sigma }_{\overline{x}}$.

28. Draw a graph of the sampling distribution of $\overline{x}$. On the number line, indicate ${\mu }_{\overline{x}},{\mu }_{\overline{x}}-3{\sigma }_{\overline{x}}$, and ${\mu }_{\overline{x}}+3{\sigma }_{\overline{x}}$.

29. In your graph, shade the area to the left of 97. Find the probability that $\overline{x}$ is less than 97.

30. In your graph of the sampling distribution of $\overline{x}$, shade the area to the left of 98. Is this area smaller or larger than the area in the previous exercise? Will the probability that $\overline{x}$ is less than 98 be greater than or less than the probability that $\overline{x}$ is less than 97? Calculate the probability that $\overline{x}$ is less than 98. Does this confirm your prediction?

31. Without using your calculator, and referring to the previous exercise, use the symmetry of the normal distribution to calculate the probability that $\overline{x}$ is greater than 102.

32. Without using your calculator, use your answers from Exercises 30 and 31 to compute the probability that $\overline{x}$ lies between 98 and 102.

33. Calculate ${\mu }_{\overline{x}}$ and $\hspace{0.17em}{\sigma }_{\overline{x}}$.

34. Draw a graph of the sampling distribution of $\overline{x}$. On the number line, indicate ${\mu }_{\overline{x}},{\mu }_{\overline{x}}-3{\sigma }_{\overline{x}}$, and ${\mu }_{\overline{x}}+3{\sigma }_{\overline{x}}$.

35. In your graph, shade the area to the right of 0.2. Find the probability that $\overline{x}$ is greater than 0.2.

36. Calculate the probability that $\overline{x}$ exceeds 0.1.

37. Without using your calculator, and referring to the previous exercise, use the symmetry of the normal distribution to calculate the probability that $\overline{x}$ is less than –0.1.

38. Without using your calculator, use your answers from Exercises 36 and 37 to compute the probability that $\overline{x}$ lies between –0.1 and 0.1.

39. Calculate ${\mu }_{\overline{x}}$ and ${\sigma }_{\overline{x}}$.

40. Draw a graph of the sampling distribution of $\overline{x}$. On the number line, indicate ${\mu }_{\overline{x}},\hspace{0.17em}{\mu }_{\overline{x}}-3{\sigma }_{\overline{x}}$, and ${\mu }_{\overline{x}}+3{\sigma }_{\overline{x}}$.

41. The value of $\overline{x}$ greater than 95% of values of $\overline{x}$

42. The value of $\overline{x}$ smaller than 95% of values of $\overline{x}$

43. The 97.5th percentile of the sample means

44. The 2.5th percentile of the sample means

45. The two symmetric values for the sample mean that contain the middle 90% of sample means. (Hint: Use your answers to Exercises 41 and 42.)

46. The two symmetric values for $\overline{x}$ that contain the middle 95% of $\overline{x}$ values. (Hint: Use your answers to Exercises 43 and 44.)

47. Calculate ${\mu }_{\overline{x}}$ and ${\sigma }_{\overline{x}}$.

48. Draw a graph of the sampling distribution of $\overline{x}$. On the number line, indicate ${\mu }_{\overline{x}},\hspace{0.17em}{\mu }_{\overline{x}}-3{\sigma }_{\overline{x}}$, and ${\mu }_{\overline{x}}+3{\sigma }_{\overline{x}}$.

49. Find the value of $\overline{x}$ that is greater than 95% of all values of $\overline{x}$.

50. Without using your calculator, use the symmetry of the normal distribution to calculate the value of $\overline{x}$ that is smaller than 95% of all values of $\overline{x}$.

51. In your graph, shade the middle 90% of the area under the curve. What are the two symmetric values for the sample mean that contain the middle 90% of sample means?

52. What proportion of $\overline{x}$ values lies outside the values you found in the previous exercise?

53. Calculate ${\mu }_{\overline{x}}$ and ${\sigma }_{\overline{x}}$.

54. Draw a graph of the sampling distribution of $\overline{x}$. On the number line, indicate ${\mu }_{\overline{x}},{\mu }_{\overline{x}}-3{\sigma }_{\overline{x}}$, and ${\mu }_{\overline{x}}+3{\sigma }_{\overline{x}}$.

55. Find the value of $\overline{x}$ that is greater than 97.5% of all values of $\overline{x}$.

56. Without using your calculator, use the symmetry of the normal distribution to calculate the value of $\overline{x}$ that is smaller than 97.5% of all values of $\overline{x}$.

57. In your graph, shade the middle 95% of the area under the curve. What are the two symmetric values for the sample mean that contain the middle 95% of sample means?

58. What proportion of $\overline{x}$ values lies outside the values you found in the previous exercise?

59. Calculate ${\mu }_{\overline{x}}$ and ${\sigma }_{\overline{x}}$.

60. Draw a graph of the sampling distribution of $\overline{x}$. On the number line, indicate ${\mu }_{\overline{x}},\hspace{0.17em}{\mu }_{\overline{x}}-3{\sigma }_{\overline{x}}$, and ${\mu }_{\overline{x}}+3{\sigma }_{\overline{x}}$.

61. In your graph, shade the area to the right of 2.6. Find the probability that $\overline{x}$ is greater than 2.6.

62. Calculate the probability that $\overline{x}$ exceeds 2.75.

63. In your graph, shade the area to the left of 2.4. Compute the probability that $\overline{x}$ is less than 2.4.

64. Now shade the area between 2.4 and 2.6. Calculate the probability that $\overline{x}$ lies between 2.4 and 2.6.

65. Calculate ${\mu }_{\overline{x}}$ and ${\sigma }_{\overline{x}}$.

66. Draw a graph of the sampling distribution of $\overline{x}$. On the number line, indicate ${\mu }_{\overline{x}},\hspace{0.17em}{\mu }_{\overline{x}}-3{\sigma }_{\overline{x}}$, and ${\mu }_{\overline{x}}+3{\sigma }_{\overline{x}}$.

67. In your graph, shade the area to the right of $-5.75$. Find the probability that $\overline{x}$ is greater than $-5.75$.

68. In your graph, shade the area to the left of $-5.25$. Calculate the probability that $\overline{x}$ is less than $-5.25$.

69. Shade the area between $-5.75$ and $-5.75$. Would you say that this represents a large probability or a small probability?

70. Calculate the probability that the sample mean golf score lies between $-5.75$ and $-5.25$.

71. Calculate ${\mu }_{\overline{x}}$ and ${\sigma }_{\overline{x}}$.

72. Draw a graph of the sampling distribution of $\overline{x}$. On the number line, indicate ${\mu }_{\overline{x}},\hspace{0.17em}{\mu }_{\overline{x}}-3{\sigma }_{\overline{x}}$, and ${\mu }_{\overline{x}}+3{\sigma }_{\overline{x}}$.

73. Find the value of $\overline{x}$ that is greater than 99.5% of all values of $\overline{x}$.

74. Without using your calculator, use the symmetry of the normal distribution to calculate the value of $\overline{x}$ that is smaller than 99.5% of all values of $\overline{x}$.

75. In your graph, shade the middle 99% of the area under the curve. What are the two symmetric values for the sample mean that contain the middle 99% of sample means?

76. What proportion of $\overline{x}$ values lies outside the values you found in the previous exercise?

77. Calculate ${\mu }_{\overline{x}}$ and ${\sigma }_{\overline{x}}$.

78. Draw a graph of the sampling distribution of $\overline{x}$. On the number line, indicate ${\mu }_{\overline{x}},\hspace{0.17em}{\mu }_{\overline{x}}-3{\sigma }_{\overline{x}}$, and ${\mu }_{\overline{x}}+3{\sigma }_{\overline{x}}$.

79. In your graph, shade the middle 95% of the area under the curve. What are the two symmetric values for the sample mean that contain the middle 95% of sample means?

80. What proportion of $\overline{x}$ values lies outside the values you found in the previous exercise?

81. The pollen count distribution for Los Angeles in September is not normally distributed, with $\mu =10$ and $\sigma =2$. Random samples of size $n=16$ are taken.

82. The pollen count distribution for Los Angeles in September is not normally distributed, with $\mu =10$ and $\sigma =2$. Random samples of size $n=64$ are taken.

83. Prices for boned trout are normally distributed, with $\mu =\$4.00$ per pound and $\sigma =\$0.50$. Random samples of size $n=16$ are taken.

84. Prices for boned trout are not normally distributed, with $\mu =\$4.00$ per pound and $\sigma =\$0.50$. Random samples of size $n=9$ are taken.

85. Accountant incomes are not normally distributed, with $\mu =\$80,000$ per year and $\sigma =\$20,000$. Random samples of size $n=10$ are taken.

86. Accountant incomes are normally distributed, with $\mu =\$80,000$ per year and $\sigma =\$20,000$. Random samples of size $n=100$ are taken.

87. Systolic blood pressure readings are not normally distributed, with $\mu =78$ and $\sigma =7$. Random samples of size $n=49$ are taken.

88. Systolic blood pressure readings are not normally distributed, with $\mu =78$ and $\sigma =7$. Random samples of size $n=25$ are taken.

89. The gas mileage for a 2014 Toyota Prius hybrid vehicle is not normally distributed, with $\mu =55$ miles per gallon and $\sigma =8$. Random samples of size $n=16$ are taken.

90. The gas mileage for a 2014 Toyota Prius hybrid vehicle is not normally distributed, with $\mu =55$ miles per gallon and $\sigma =8$. Random samples of size $n=64$ are taken.

91. Blood Lead Contamination. A study² of workers in lead-exposed jobs found that the mean blood lead level was $\mu =31.4$ micrograms, with a standard deviation of $\sigma =14.2$ micrograms. Let the sample size be $n=16$. Assume the distribution is normal.

92. Student Debt. The Project on Student Debt published a study³ in which they found that the mean amount of student debt owed by American college students was $\mu =\$29,400$. Assume a standard deviation of $\sigma =\$9000$. Suppose we take a sample of $n=9$ students. Assume the distribution is normal.

93. Measuring Well-Being. The Gallup-Healthways Well-Being Index represents an average of the reported emotional health, physical health, healthy behavior, and work environment of the survey respondents.⁴ Gallup reported that the mean Well-Being Index in 2013 was $\mu =66.2$. Assume the standard deviation was $\sigma =10$. Suppose we take a sample of 25 survey respondents. If we assume the underlying distribution is normal, find the probability that the sample mean Well-Being Index will have the following values:

94. Statisticians' Salaries. If you enjoy statistics, you may want to become a statistician. The U.S. Census Bureau reported in 2014 that the median salary for mathematicians and statisticians was about $86,000. Income data is usually right-skewed, so we may safely assume that the mean is higher. Suppose the salaries of statisticians have a mean of $\mu =\$100,000$ and a standard deviation of $\sigma =\$20,000$.

Suppose we take samples of size 100 statisticians. Find the probability that the sample mean salary will have the following values:

95. Blood Lead Contamination. Refer to Exercise 91.

96. Student Debt. Refer to Exercise 92.

97. Measuring Well-Being. Refer to Exercise 93.

98. Statisticians' Salaries. Refer to Exercise 94.

99. Cholesterol Levels. The Centers for Disease Control and Prevention reports that the mean serum cholesterol level in Americans is 202. Assume that the standard deviation is 45. There is no information about the distribution. We take a sample of 36 Americans.

100. Tennessee Temperatures. According to the National Oceanic and Atmospheric Administration, the mean temperature for Nashville, Tennessee, in the month of January between 1872 and 2014 was 38.6°F. Assume that the standard deviation is 10°F, but the distribution is unknown. If we take a sample of $n=36$, find the following probabilities:

101. Computers per School. The National Center for Educational Statistics (http://nces.ed.gov) reported that the mean number of instructional computers per public school nationwide was 124. Assume that the standard deviation is 50 computers and no information is available about the shape of the distribution. Suppose we take a sample of size 100 public schools. Compute the following probabilities:

102. Cholesterol Levels. Refer to Exercise 99.

103. Tennessee Temperatures. Refer to Exercise 100.

104. Computers per School. Refer to Exercise 101.

105. Carbon Dioxide Concentration. The Carbon Dioxide Information Analysis Center (cdiac.ornl.gov/pns/current_ghg.html) reported in 2014 that the mean concentration of the greenhouse gas carbon dioxide is 397 parts per million (ppm). Suppose the population is normally distributed, assume that the standard deviation is 25 ppm, and suppose we take atmospheric samples from 25 cities.

106. Short-Term Memory. In a famous research paper in the psychology literature, George Miller found that the amount of information humans could process in short-term memory was 7 bits (pieces of information), plus or minus 2 bits.⁵ Assume that the mean number of bits is 7 and the standard deviation is 2, and that the distribution is normal. Suppose we take a sample of 100 people and test their short-term memory skills.

107. Does the normal probability plot show evidence in favor of normality or against normality? What characteristics of the plot illustrate this evidence?

108. If possible, find the probability that a random sample of $n=9$ patients will have a mean blood glucose level greater than 86. If not possible, explain why not.

109. If possible, find the probability that a random sample of $n=36$ patients will have a mean blood glucose level greater than 86. If not possible, explain why not.

110. Does the normal probability plot show evidence in favor of normality or against normality? What characteristics of the plot illustrate this evidence?

111. If possible, find the probability that a random sample of $n=16$ instructors will have a mean adjusted gross income between $40,000 and $60,000. If not possible, explain why not.

112. If possible, find the probability that a random sample of $n=36$ instructors will have a mean adjusted gross income between $40,000 and $60,000. If not possible, explain why not.

113. Refer to Exercise 112. What if the sample size used was some unspecified value greater than 36? Describe how and why this change would have affected the following, if at all. Would the quantities increase, decrease, remain unchanged? Or is there insufficient information to tell what would happen? Explain your answers.

114. What were the chances that the Master of the Mint would have been accused of cheating if he had in fact been completely honest?

115. Use the methods shown in the Case Study to answer the following questions:

116. Assume “just a little debasement” from 128 grams per coin to 127.9 grams per coin.

117. Assume that greed attacked the Master of the Mint and he went for the big debasement from 128 grams per coin to 127.3 grams per coin.

118. The Master of the Mint wanted to choose a mean value so that his chances of failing the Trial of the Pyx were 25%. Which mean value is this?

119. Describe the shape of the sampling distribution of $\overline{x}$ for the following sample sizes:

120. At what sample size would you say the sampling distribution of $\overline{x}$ becomes approximately normal?

121. What is the probability that a randomly selected SAT Math score will be less than 500?

122. As a researcher, you are looking at samples of SAT Math scores of size 16.

123. Refer to Exercise 122.

124. Calculate the following sample mean values:

125. Refer to Exercise 123. What if the population standard deviation was greater than 116? Explain how this would affect the following, if at all:

126. What if the population standard deviation was greater than 116? Explain how this would affect the following, if at all: