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CHAPTER 9 SECTION 1: SAMPLING DISTRIBUTIONS MULTIPLE CHOICE 1. The standard deviation of the sampling distribution of is also called the: a. central limit theorem. b. standard error of the sample mean. c. finite population correction factor. d. population standard deviation. ANS: B PTS: 1 REF: SECTION 9.1 2. Random samples of size 49 are taken from an infinite population whose mean is 300 and standard deviation is 21. The mean and standard error of the sample mean, respectively, are: a. 300 and 21 b. 300 and 3 c. 300 and 0.43 d. None of these choices. ANS: B PTS: 1 REF: SECTION 9.1 3. Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, is less than 78 is: a. 0.9332 b. 0.5987 c. 1.5000 d. None of these choices. ANS: A PTS: 1 REF: SECTION 9.1 4. An infinite population has a mean of 40 and a standard deviation of 15. A sample of size 100 is taken at random from this population. The standard error of the sample mean equals: a. 15 b. 15/100 c. 15/100 d. None of these choices. ANS: C PTS: 1 REF: SECTION 9.1 5. If all possible samples of size n are drawn from an infinite population with a mean of 15 and a standard deviation of 5, then the standard error of the sample mean equals 1.0 for samples of size: a. 5 b. 15 c. 25 d. None of these choices. ANS: C PTS: 1 REF: SECTION 9.1 6. As a general rule in computing the standard error of the sample mean, the finite population correction factor is used only if the: a. sample size is smaller than 5% of the population size. b. sample size is greater than 5% of the sample size. c. sample size is more than half of the population size. d. None of these choices. ANS: B PTS: 1 REF: SECTION 9.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 7. Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals: a. 25 b. 25/64 c. 25/64 d. None of these choices. ANS: C PTS: 1 REF: SECTION 9.1 8. A sample of size 40 is taken from an infinite population whose mean and standard deviation are 68 and 12, respectively. The probability that the sample mean is larger than 70 equals a. P(Z > 70) b. P(Z > 2) c. P(Z > 0.17) d. P(Z > 1.05) ANS: D PTS: 1 REF: SECTION 9.1 9. The finite population correction factor should be used: a. whenever we are sampling from an infinite population. b. whenever we are sampling from a finite population. c. whenever the sample size is large compared to the population size. d. whenever the sample size is small compared to the population size. ANS: C PTS: 1 REF: SECTION 9.1 10. Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 45 and 9, respectively. The mean and standard error of the sampling distribution of the sample mean are: a. 45 and 9 b. 45/81 and 9/81 c. 45 and 9/81 d. 45/81 and 9/81 ANS: C PTS: 1 REF: SECTION 9.1 11. A sample of size 25 is selected at random from a finite population. If the finite population correction factor is 0.63, then the population size is: a. 25 b. 66 c. 41 d. None of these choices. ANS: C PTS: 1 REF: SECTION 9.1 12. The Central Limit Theorem states that, if a random sample of size n is drawn from a population, then the sampling distribution of the sample mean : a. is approximately normal if n > 30. b. is approximately normal if n < 30. c. is approximately normal if the underlying population is normal. d. None of these choices. ANS: A PTS: 1 REF: SECTION 9.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 13. A sample of size n is selected at random from an infinite population. As n increases, which of the following statements is true? a. The population standard deviation decreases. b. The standard error of the sample mean decreases. c. The population standard deviation increases. d. The standard error of the sample mean increases. ANS: B PTS: 1 REF: SECTION 9.1 14. The expected value of the sampling distribution of the sample mean a. only when the population is normally distributed. b. only when the sample size is large. c. only when the population is infinite. d. for all populations. ANS: D PTS: 1 equals the population mean : REF: SECTION 9.1 15. If all possible samples of size n are drawn from an infinite population with a mean of and a standard deviation of , then the standard error of the sample mean is inversely proportional to: a. b. c. n d. ANS: D PTS: 1 REF: SECTION 9.1 16. If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean will be: a. normal for all values of n. b. normal only for n > 30. c. approximately normal for all values of n. d. approximately normal only for n > 30. ANS: A PTS: 1 REF: SECTION 9.1 17. If all possible samples of size n are drawn from a population, the probability distribution of the sample mean is called the: a. standard error of . b. expected value of . c. sampling distribution of . d. normal distribution. ANS: C PTS: 1 REF: SECTION 9.1 18. Sampling distributions describe the distributions of: a. population parameters. b. sample statistics. c. both parameters and statistics d. None of these choices. ANS: B PTS: 1 REF: SECTION 9.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 19. Suppose X has a distribution that is not normal. The Central Limit Theorem is important in this case because: a. it says the sampling distribution of is approximately normal for any sample size. b. it says the sampling distribution of is approximately normal if n is large enough. c. it says the sampling distribution of is exactly normal, for any sample size. d. None of these choices. ANS: B PTS: 1 REF: SECTION 9.1 20. Which of the following statements about the sampling distribution of is NOT true? a. It is generated by taking all possible samples of size n and computing their sample means. b. Its mean is equal to the population mean . c. Its standard deviation is equal to the population standard deviation . d. All of these choices are true. ANS: C PTS: 1 REF: SECTION 9.1 21. The standard error of the mean: a. is never larger than the standard deviation of the population. b. decreases as the sample size increases. c. measures the variability of the mean from sample to sample. d. All of these choices are true. ANS: D PTS: 1 REF: SECTION 9.1 22. Which of the following is true about the sampling distribution of the sample mean? a. Its mean is always equal to b. Its standard error is always equal to the population standard deviation . c. Its shape is exactly normal if n is large enough. d. None of these choices. ANS: D PTS: 1 REF: SECTION 9.1 23. The owner of a fish market has an assistant who has determined that the weights of catfish are normally distributed, with a mean of 3.2 pounds and standard deviation of 0.8 pounds. If a sample of 25 fish yields a mean of 3.6 pounds, what is the Z-score for this sample mean? a. 2.50 b. 2.50 c. 0.50 d. None of these choices. ANS: A PTS: 1 REF: SECTION 9.1 24. The owner of a fish market has an assistant who has determined that the weights of catfish are normally distributed, with a mean of 3.2 pounds and standard deviation of 0.84 pounds. If a sample of 16 fish is taken, what is the standard error of the mean weight? a. 0.840 b. 0.053 c. 0.210 d. None of these choices. ANS: C PTS: 1 REF: SECTION 9.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 25. Suppose the ages of students in your university or college follow a positively skewed distribution with mean of 24 years and a standard deviation of 4 years. If we randomly sampled 100 students, which of the following statements about the sampling distribution of the sample mean age is NOT true? a. The mean of the sampling distribution of sample mean is equal to 24 years. b. The standard deviation of the sampling distribution of sample mean is equal to 4 years. c. The shape of the sampling distribution of sample mean is approximately normal. d. All of these choices are true. ANS: B PTS: 1 REF: SECTION 9.1 26. Suppose that 100 items are drawn from a population of manufactured products and the weight, X, of each item is recorded. Prior experience has shown that the weight has a non-normal probability distribution with = 8 ounces and = 3 ounces. Which of the following is true about the sampling of ? a. Its mean is 8 ounces. b. Its standard error is 0.3 ounces. c. Its shape is approximately normal. d. All of these choices are true. ANS: D PTS: 1 REF: SECTION 9.1 27. The standard error of the mean for a sample of 100 is 25. In order to cut the standard error of the mean in half (to 12.5) we must: a. increase the sample size to 200. b. decrease the sample size to 50. c. keep the sample size at 100 and change something else. d. None of these choices. ANS: D PTS: 1 REF: SECTION 9.1 28. Which of the following is true regarding the sampling distribution of the mean for a large sample size? Assume the population distribution is not normal. a. It has the same shape, mean and standard deviation as the population. b. It has the same shape and mean as the population, but a different standard deviation. c. It the same mean and standard deviation as the population, but a different shape. d. It has the same mean as the population, but a different shape and standard deviation. ANS: D PTS: 1 REF: SECTION 9.1 29. For sample sizes greater than 30, the sampling distribution of the mean is approximately normally distributed: a. regardless of the shape of the population. b. only if the shape of the population is symmetric. c. only if the population is normally distributed. d. None of these choices. ANS: A PTS: 1 REF: SECTION 9.1 30. For a sample size of 1, the sampling distribution of the mean is normally distributed: a. regardless of the shape of the population. b. only if the population values are larger than 30. c. only if the population is normally distributed. d. None of these choices. ANS: C PTS: 1 REF: SECTION 9.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. TRUE/FALSE 31. When a great many simple random samples of size n are drawn from a population that is normally distributed, the sampling distribution of the sample mean is normal regardless of the sample size n. ANS: T PTS: 1 REF: SECTION 9.1 32. The Central Limit Theorem permits us to draw conclusions about a population based on a sample alone, without having any knowledge about the distribution of that population. And this works no matter what the sample size is. ANS: F PTS: 1 REF: SECTION 9.1 33. Consider an infinite population with a mean of 100 and a standard deviation of 20. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals 2.50. ANS: T PTS: 1 REF: SECTION 9.1 34. If all possible samples of size n are drawn from an infinite population with standard deviation 8, then the standard error of the sample mean equals 1.0 if the sample size is 64. ANS: T PTS: 1 REF: SECTION 9.1 35. A sample of size n is selected at random from an infinite population. As n increases, the standard error of the sample mean increases. ANS: F PTS: 1 REF: SECTION 9.1 36. A sample of size 25 is selected from a population of size 500. The finite population correction is needed to find the standard error of . ANS: F PTS: 1 REF: SECTION 9.1 37. A sample of size 25 is selected from a population of size 75. The finite population correction needed to find the standard error of . ANS: T PTS: 1 REF: SECTION 9.1 38. The amount of time it takes to complete a final examination is negatively skewed distribution with a mean of 70 minutes and a standard deviation of 8 minutes. If 64 students were randomly sampled, the probability that the sample mean of the sampled students exceeds 73.5 minutes is approximately 0. ANS: T PTS: 1 REF: SECTION 9.1 39. If all possible samples of size n are drawn from a normal population, the probability distribution of the sample mean is an exact normal distribution. ANS: T PTS: 1 REF: SECTION 9.1 40. If the sample size increases, the standard error of the mean also increases. ANS: F PTS: 1 REF: SECTION 9.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 41. The amount of bleach a machine pours into bottles has a mean of 50 ounces with a standard deviation of 0.25 ounces. We take a random sample of 36 bottles filled by this machine. The sampling distribution of the sample mean has a mean of 50 ounces. ANS: T PTS: 1 REF: SECTION 9.1 42. If the population distribution is skewed, in most cases the sampling distribution of the sample mean can be approximated by the normal distribution if the samples contain at least 30 observations. ANS: T PTS: 1 REF: SECTION 9.1 43. A sampling distribution is a probability distribution for a statistic, not a parameter. ANS: T PTS: 1 REF: SECTION 9.1 44. A sampling distribution is defined as the probability distribution of means from all possible sample sizes that are taken from a given population. ANS: F PTS: 1 REF: SECTION 9.1 45. If the population distribution is unknown, in most cases the sampling distribution of the mean can be approximated by the normal distribution if the samples contain at least 30 observations. ANS: T PTS: 1 REF: SECTION 9.1 46. The amount of bleach a machine pours into bottles has a mean of 50 ounces with a standard deviation of 0.25 ounces. Suppose we take a random sample of 36 bottles filled by this machine. The sampling distribution of the sample mean has a standard error of 0.25 ounces. ANS: F PTS: 1 REF: SECTION 9.1 47. As the size of the sample is increased, the standard error of ANS: T PTS: 1 REF: SECTION 9.1 48. As the size of the sample is increased, the mean of ANS: F PTS: 1 decreases. increases. REF: SECTION 9.1 49. In inferential statistics, the standard error of the sample mean assesses the uncertainty or error of estimation. ANS: T PTS: 1 REF: SECTION 9.1 COMPLETION 50. Because the value of the ____________________ varies from sample to sample, we can regard it as a new random variable, created by sampling. ANS: sample mean PTS: 1 REF: SECTION 9.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 51. The variance of the sampling distribution of is ____________________ the variance of the population we're sampling from, for any n > 1. ANS: less than PTS: 1 REF: SECTION 9.1 52. A randomly selected value of is likely to be ____________________ to(than) the mean of the population than is a randomly selected value of X. ANS: closer PTS: 1 REF: SECTION 9.1 53. As the number of throws of a fair die increases, the probability that the sample mean is close to (the number) ____________________ increases. ANS: 3.5 PTS: 1 REF: SECTION 9.1 54. The width of the sampling distribution of gets ____________________ as the sample size increases. ANS: narrower more narrow PTS: 1 REF: SECTION 9.1 55. As n gets ____________________, the shape of the sampling distribution of bell shaped. becomes increasingly ANS: larger PTS: 1 REF: SECTION 9.1 56. As n gets larger, the sampling distribution of due to the ____________________. becomes increasingly bell shaped. This phenomenon is ANS: Central Limit Theorem PTS: 1 REF: SECTION 9.1 57. The accuracy of the approximation of with a normal distribution depends on the probability distribution of the ____________________ and on the sample ____________________. ANS: population; size PTS: 1 REF: SECTION 9.1 58. If the population is normal, then is normally distributed for __________________ values of n. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. ANS: all PTS: 1 REF: SECTION 9.1 59. The finite population correction is not needed if the population size is large relative to the sample size. As a rule of thumb, we will treat any population that is at least ____________________ times larger than the sample size as large. ANS: 20 twenty PTS: 1 REF: SECTION 9.1 SHORT ANSWER Children Heights Heights of 10-year-old children are normally distributed with a mean of 52 inches and a standard deviation of 4 inches. 60. {Children Heights Narrative} Find the probability that one randomly selected 10-year-old child is under 54 inches. ANS: 0.6915 PTS: 1 REF: SECTION 9.1 61. {Children Heights Narrative} Find the probability that two randomly selected 10-year-old children are both under 54 inches. ANS: (0.6915)(0.6915) = 0.4782 PTS: 1 REF: SECTION 9.1 62. {Children Heights Narrative} Find the probability that the mean height of 4 randomly selected 10year-old children is under 54 inches. ANS: 0.8413 PTS: 1 REF: SECTION 9.1 Average Annual Income Suppose that the average annual income of a defense attorney is $150,000 with a standard deviation of $40,000. Assume that the income distribution is normal. 63. {Average Annual Income Narrative} What is the probability that one defense attorney selected at random makes less than $120,000? This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. ANS: 0.2266 PTS: 1 REF: SECTION 9.1 64. {Average Annual Income Narrative} What is the probability that the average annual income of a random sample of 4 defense attorneys is less than $120,000? ANS: 0.0668 PTS: 1 REF: SECTION 9.1 65. {Average Annual Income Narrative} Why are your answers to the previous two questions different? ANS: The salary of one attorney chosen from the population has more variability than the average salary of 4 attorneys. That means that a salary lower than 120,000 is harder to achieve with an average of 4 individuals, compared to one individual. PTS: 1 REF: SECTION 9.1 66. {Average Annual Income Narrative} Could you have used a normal distribution to find an (approximate) probability for the average of 4 salaries if the population of salaries did not have a normal distribution? ANS: No, because the sample size of 4 is too small for the Central Limit Theorem to be used. PTS: 1 REF: SECTION 9.1 Mean Salary In order to estimate the mean salary for a population of 500 employees, the president of a certain company selected at random a sample of 40 employees. 67. {Mean Salary Narrative} Would you use the finite population correction factor in calculating the standard error of the sample mean in this case? Explain. ANS: Since the population size is 12.5 times as large as the sample size (500/40 = 12.5), the finite population correction factor is necessary. (It doesn't have to be used if the population is at least 20 times as large as the sample.) PTS: 1 REF: SECTION 9.1 68. {Mean Salary Narrative} If the population standard deviation is $800, compute the standard error both with and without using the finite population correction factor. ANS: = $121.448 and $126.491 with and without the finite population correction factor, respectively. The finite population correction factor does make a difference. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. PTS: 1 REF: SECTION 9.1 Senior Citizens A sample of 50 senior citizens is drawn at random from a normal population whose mean age and standard deviation are 75 and 6 years, respectively. 69. {Senior Citizens Narrative} Describe the shape of the sampling distribution of the sample mean in this case. ANS: is normal because the original population is normal. This is true for any sample size. PTS: 1 REF: SECTION 9.1 70. {Senior Citizens Narrative} Find the mean and standard error of the sampling distribution of the sample mean. ANS: = 75 and = .8485 PTS: 1 REF: SECTION 9.1 71. {Senior Citizens Narrative} What is the probability that the mean age exceeds 73 years? ANS: P( > 73) = 0.9909 PTS: 1 REF: SECTION 9.1 72. {Senior Citizens Narrative} What is the probability that the mean age is at most 73 years? ANS: P( 73) = 0.0091 PTS: 1 REF: SECTION 9.1 73. {Senior Citizens Narrative} What is the probability that two randomly selected seniors are over 73 years of age? ANS: 0.9909 0.9909 = 0.9819 PTS: 1 REF: SECTION 9.1 Heights of Men The heights of men in the USA are normally distributed with a mean of 68 inches and a standard deviation of 4 inches. 74. {Heights of Men Narrative} What is the probability that a randomly selected man is taller than 70 inches? This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. ANS: 0.3085 PTS: 1 REF: SECTION 9.1 75. {Heights of Men Narrative} A random sample of five men is selected. What is the probability that the sample mean is greater than 70 inches? ANS: 0.1314 PTS: 1 REF: SECTION 9.1 76. {Heights of Men Narrative} What is the probability that the mean height of a random sample of 36 men is greater than 70 inches? ANS: 0.0013 PTS: 1 REF: SECTION 9.1 77. {Heights of Men Narrative} If the population of men's heights is not normally distributed, which, if any, of the previous questions can you answer? ANS: If heights were not normal, we could only answer the question about the mean height of 36 men. PTS: 1 REF: SECTION 9.1 Sports Time The amount of time spent by American adults playing sports per week is normally distributed with a mean of 4 hours and standard deviation of 1.25 hours. 78. {Sports Time Narrative} Find the probability that a randomly selected American adult plays sports for more than 5 hours per week. ANS: 0.2119 PTS: 1 REF: SECTION 9.1 79. {Sports Time Narrative} Find the probability that if four American adults are randomly selected, their average number of hours spent playing sports is more than 5 hours per week. ANS: 0.0548 PTS: 1 REF: SECTION 9.1 80. {Sports Time Narrative} Find the probability that if four American adults are randomly selected, all four play sports for more than 5 hours per week. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. ANS: (0.2119)4 = 0.0020 PTS: 1 REF: SECTION 9.1 Number of Pets The following data give the number of pets owned for a population of 4 families. Family Number of Pets Owned A 2 B 1 C 4 D 3 81. {Number of Pets Narrative} Find the mean and the standard deviation for the population. ANS: = 2.5 pets and = 1.12 pets PTS: 1 REF: SECTION 9.1 82. {Number of Pets Narrative} A sample of size 2 is drawn at random from the population. Use the formulas and to calculate the mean and the standard deviation of the sampling distribution of the sample means. ANS: and PTS: 1 REF: SECTION 9.1 83. {Number of Pets Narrative} List all possible samples of 2 families that can be selected without replacement from this population, and compute the sample mean for each sample. ANS: Sample A, B 1.5 PTS: 1 A, C 3.0 A, D 2.5 B, C 2.5 B, D 2.0 C, D 3.5 REF: SECTION 9.1 84. {Number of Pets Narrative} Find the sampling distribution of , and use it to calculate the mean and the standard deviation of . ANS: p( ) 1.5 1/6 2.0 1/6 2.5 1/6 3.0 1/6 3.5 1/6 , and This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. PTS: 1 REF: SECTION 9.1 Newspapers A local newspaper sells an average of 2,100 papers per day, with a standard deviation of 500 papers. Consider a sample of 60 days of operation. 85. {Newspapers Narrative} What is the shape of the sampling distribution of the sample mean number of papers sold per day? Why? ANS: Approximately normal since n > 30. PTS: 1 REF: SECTION 9.1 86. {Newspapers Narrative} Find the expected value and the standard error of the sample mean. ANS: = 2,100 papers per day and PTS: 1 = 65.55 papers per day REF: SECTION 9.1 87. {Newspapers Narrative} What is the probability that the sample mean is between 2,000 and 2,300 papers? ANS: 0.9384 PTS: 1 REF: SECTION 9.1 88. In a given year, the average annual salary of a NFL football player was $205,000 with a standard deviation of $24,500. If a simple random sample of 50 players was taken, what is the probability that the sample mean will exceed $210,000? ANS: 0.0749 PTS: 1 REF: SECTION 9.1 89. An auditor knows from past history that the average accounts receivable for a company is $521.72 with a standard deviation of $584.64. If the auditor takes a simple random sample of 100 accounts, what is the probability that the mean of the sample is within $120 of the population mean? ANS: 0.9596 PTS: 1 REF: SECTION 9.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher.