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CHAPTER 8 SECTION 2: CONTINUOUS PROBABILITY DISTRIBUTIONS MULTIPLE CHOICE 55. A standard normal distribution is a normal distribution with: a. a mean of zero and a standard deviation of one. b. a mean of one and a standard deviation of zero. c. a mean always larger than the standard deviation. d. None of these choices. ANS: A PTS: 1 REF: SECTION 8.2 56. What proportion of the data from a normal distribution is within two standard deviations from the mean? a. 0.3413 b. 0.4772 c. 0.6826 d. 0.9544 ANS: D PTS: 1 REF: SECTION 8.2 57. Given that Z is a standard normal random variable, the area to the left of a value z is expressed as a. P(Z z) b. P(Z z) c. P(0 Z z) d. P(Z z) ANS: B PTS: 1 REF: SECTION 8.2 58. If X has a normal distribution with mean 60 and standard deviation 6, which value of X corresponds with the value z = 1.96? a. x = 71.76 b. x = 67.96 c. x = 61.96 d. x = 48.24 ANS: A PTS: 1 REF: SECTION 8.2 59. Which of the following is not a characteristic for a normal distribution? a. It is symmetrical. b. The mean is always zero. c. The mean, median, and mode are all equal. d. It is a bell-shaped distribution. ANS: B PTS: 1 REF: SECTION 8.2 60. Given that Z is a standard normal variable, the variance of Z: a. is always greater than 2.0. b. is always greater than 1.0. c. is always equal to 1.0. d. cannot assume a specific value. ANS: C PTS: 1 REF: SECTION 8.2 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. 61. Given that Z is a standard normal random variable, a negative value (z) on its distribution would indicate: a. z is to the left of the mean. b. the standard deviation of this Z distribution is negative. c. the area between zero and the value z is negative. d. None of these choices. ANS: A PTS: 1 REF: SECTION 8.2 62. A larger standard deviation of a normal distribution indicates that the distribution becomes: a. narrower and more peaked. b. flatter and wider. c. more skewed to the right. d. more skewed to the left. ANS: B PTS: 1 REF: SECTION 8.2 63. In its standardized form, the normal distribution: a. has a mean of 0 and a standard deviation of 1. b. has a mean of 1 and a variance of 0. c. has an area equal to 0.5. d. cannot be used to approximate discrete probability distributions. ANS: A PTS: 1 REF: SECTION 8.2 64. Most values of a standard normal distribution lie between: a. 0 and 1 b. 3 and 3 c. 0 and 3 d. minus infinity and plus infinity ANS: B PTS: 1 REF: SECTION 8.2 65. Bob took a math test whose mean was 70 and standard deviation was 5. The total points possible was 100. Bob's results were reported to be at the 95th percentile. What was Bob's actual exam score, rounded to the nearest whole number? a. 95 b. 78 c. 75 d. 62 ANS: B PTS: 1 REF: SECTION 8.2 66. Bob took a statistics test whose mean was 80 and standard deviation was 5. The total points possible was 100. Bob's score was 2 standard deviations below the mean. What was Bob's score, rounded to the nearest whole number? a. 78 b. 70 c. 90 d. None of these choices. ANS: B PTS: 1 REF: SECTION 8.2 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. 67. Bob took a biology exam whose mean was 70 with standard deviation 5. He also took a chemistry exam whose mean was 80 with standard deviation 10. He scored 85 on both exams. On which exam did he do better compared to the other students who took the exam? a. He did better on the biology exam, comparatively speaking. b. He did better on the chemistry exam, comparatively speaking. c. He did the same on both exams, relatively speaking. d. Cannot tell without more information. ANS: A PTS: 1 REF: SECTION 8.2 68. Suppose Bob's exam score was at the 80th percentile on an exam whose mean was 90. What was Bob's exam score? a. 76.81 b. 72.00 c. 80.00 d. Cannot tell without more information. ANS: D PTS: 1 REF: SECTION 8.2 TRUE/FALSE 69. If we standardize the normal curve, we express the original X values in terms of their number of standard deviations away from the mean. ANS: T PTS: 1 REF: SECTION 8.2 70. A normal distribution is symmetric; therefore the probability of being below the mean is 0.50 and the probability of being above the mean is 0.50. ANS: T PTS: 1 REF: SECTION 8.2 71. A random variable X is standardized by subtracting the mean and dividing by the variance. ANS: F PTS: 1 REF: SECTION 8.2 72. A random variable X has a normal distribution with mean 132 and variance 36. If x = 120, its corresponding value of Z is 2.0. ANS: F PTS: 1 REF: SECTION 8.2 73. A random variable X has a normal distribution with a mean of 250 and a standard deviation of 50. Given that X = 175, its corresponding value of Z is 1.50. ANS: T PTS: 1 REF: SECTION 8.2 74. Given that Z is a standard normal random variable, a negative value of Z indicates that the standard deviation of Z is negative. ANS: F PTS: 1 REF: SECTION 8.2 75. In the standard normal distribution, z0.05 = 1.645 means that 5% of all values of z are below 1.645 and 95% are above it. ANS: F PTS: 1 REF: SECTION 8.2 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. 76. The probability that a standard normal random variable Z is less than 3.5 is approximately 0. ANS: T PTS: 1 REF: SECTION 8.2 77. If the value of Z is z = 99, that means you are at the 99th percentile on the Z distribution. ANS: F PTS: 1 REF: SECTION 8.2 78. The 10th percentile of a Z distribution has 10% of the Z-values lying above it. ANS: F PTS: 1 REF: SECTION 8.2 79. The probability that Z is less than 2 is the same as one minus the probability that Z is greater than +2. ANS: F PTS: 1 REF: SECTION 8.2 80. If your golf score is 3 standard deviations below the mean, its corresponding value on the Z distribution is 3. ANS: T PTS: 1 REF: SECTION 8.2 81. Suppose X has a normal distribution with mean 70 and standard deviation 5. The 50th percentile of X is 70. ANS: T PTS: 1 REF: SECTION 8.2 82. A national standardized testing company can tell you your relative standing on an exam without divulging the mean or the standard deviation of the exam scores. ANS: T PTS: 1 REF: SECTION 8.2 COMPLETION 83. Suppose X has a normal distribution with mean 40 and standard deviation 2. Shifting all the X values to the right 10 units results in a normal distribution with mean ____________________ and standard deviation ____________________. ANS: 50; 2 fifty; two PTS: 1 REF: SECTION 8.2 84. ____________________ the value of in a normal distribution will make it wider. ANS: Increasing PTS: 1 REF: SECTION 8.2 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. 85. We standardize a random variable by subtracting its ____________________ and dividing by its ____________________. ANS: mean; standard deviation PTS: 1 REF: SECTION 8.2 86. Suppose X has a normal distribution with mean 10 and standard deviation 2. The probability that X is less than 8 is equal to the probability that Z is less than ____________________. ANS: 1 PTS: 1 REF: SECTION 8.2 87. P(Z > 1.9) = ____________________ P(Z < 1.9). ANS: 1 PTS: 1 REF: SECTION 8.2 88. P(1 < Z < 2) = P(Z < 2) ____________________. ANS: P(Z < 1) P(Z<1) PTS: 1 REF: SECTION 8.2 89. The mean of the standard normal distribution is ____________________ and the standard deviation is ____________________. ANS: 0; 1 zero; one PTS: 1 REF: SECTION 8.2 90. P(Z > 3.00) is approximately ____________________. ANS: 0 zero PTS: 1 REF: SECTION 8.2 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. 91. P(Z < 3.00) is approximately ____________________. ANS: 1 one PTS: 1 REF: SECTION 8.2 92. Suppose X is a normal random variable with mean 70 and standard deviation 3. Then P(X = 3) = ____________________. ANS: 0 zero PTS: 1 REF: SECTION 8.2 93. Z.025 is the value of Z such that the area to the ____________________ of Z is .9750. ANS: left PTS: 1 REF: SECTION 8.2 SHORT ANSWER Lamps Lifetime A certain brand of flood lamps has a lifetime that has a normal distribution with a mean of 3,750 hours and a standard deviation of 300 hours. 94. {Lamps Lifetime Narrative} What proportion of these lamps will last for more than 4,000 hours? ANS: 0.2033 PTS: 1 REF: SECTION 8.2 95. {Lamps Lifetime Narrative} What proportion of these lamps will last less than 3,600 hours? ANS: 0.3085 PTS: 1 REF: SECTION 8.2 96. {Lamps Lifetime Narrative} What proportion of these lamps will last between 3,800 and 4,100 hours? ANS: 0.3115 PTS: 1 REF: SECTION 8.2 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. 97. {Lamps Lifetime Narrative} What lifetime should the manufacturer advertise for these lamps in order that only 2% of the lamps will burn out before the advertised lifetime? ANS: 3135 PTS: 1 REF: SECTION 8.2 Diet Researchers studying the effects of a new diet found that the weight loss over a one-month period by those on the diet was normally distributed with a mean of 10 pounds and a standard deviation of 5 pounds. 98. {Diet Narrative} What proportion of the dieters lost more than 12 pounds? ANS: 0.3446 PTS: 1 REF: SECTION 8.2 99. {Diet Narrative} What proportion of the dieters gained weight? ANS: 0.0028 PTS: 1 REF: SECTION 8.2 100. {Diet Narrative} If a dieter is selected at random, what is the probability that the dieter lost more than 7.5 pounds? ANS: 0.6915 PTS: 1 REF: SECTION 8.2 101. Let X be a normally distributed random variable with a mean of 12 and a standard deviation of 1.5. What proportions of the values of X are: a. b. c. less than 14 more than 8 between 10 and 13 ANS: a. b. c. 0.9082 0.9962 0.6568 PTS: 1 REF: SECTION 8.2 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. 102. If Z is a standard normal random variable, find the value z for which: a. b. c. d. the area between 0 and z is 0.3729 the area to the right of z is 0.7123 the area to the left of z is 0.1736 the area between z and z is 0.6630 ANS: a. b. c. d. 1.14 .56 .94 0.96 PTS: 1 REF: SECTION 8.2 103. If Z is a standard normal random variable, find the following probabilities: a. b. c. d. e. P(Z 1.77) P(Z 1.96) P(0.35 Z 0.85) P(2.88 Z 2.15) P(Z 1.45) ANS: a. b. c. d. e. 0.0384 0.9750 0.1655 0.0138 0.9265 PTS: 1 REF: SECTION 8.2 Math Scores Scores of high school students on a national mathematics exam were normally distributed with a mean of 86 and a standard deviation of 4. (Total possible points = 100.) 104. {Math Scores Narrative} What is the probability that a randomly selected student will have a score of 80 or higher? ANS: 0.9332 PTS: 1 REF: SECTION 8.2 105. {Math Scores Narrative} What is the probability that a randomly selected student will have a score between 80 and 90? 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: 0.7745 PTS: 1 REF: SECTION 8.2 106. {Math Scores Narrative} What is the probability that a randomly selected student will have a score of 94 or lower? ANS: 0.9772 PTS: 1 REF: SECTION 8.2 Saving Accounts A bank has determined that the monthly balances of the saving accounts of its customers are normally distributed with an average balance of $1,200 and a standard deviation of $250. 107. {Saving Accounts Narrative} What proportion of customers have monthly balances less than $1,000? ANS: 0.2119 PTS: 1 REF: SECTION 8.2 108. {Saving Accounts Narrative} What proportion of customers have monthly balances more than $1,125? ANS: 0.6179 PTS: 1 REF: SECTION 8.2 109. {Saving Accounts Narrative} What proportion of customers have monthly balances between $950 and $1,075? ANS: 0.1498 PTS: 1 REF: SECTION 8.2 CIS Graduate Salary The recent average starting salary for new college graduates in computer information systems is $47,500. Assume salaries are normally distributed with a standard deviation of $4,500. 110. {CIS Graduate Salary Narrative} What is the probability of a new graduate receiving a salary between $45,000 and $50,000? ANS: 0.4246 PTS: 1 REF: SECTION 8.2 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. 111. {CIS Graduate Salary Narrative} What is the probability of a new graduate getting a starting salary in excess of $55,000? ANS: 0.0475 PTS: 1 REF: SECTION 8.2 112. {CIS Graduate Salary Narrative} What percent of starting salaries are no more than $42,250? ANS: 12.10% PTS: 1 REF: SECTION 8.2 113. {CIS Graduate Salary Narrative} What is the cutoff for the bottom 5% of the salaries? ANS: $40,097.50 PTS: 1 REF: SECTION 8.2 114. {CIS Graduate Salary Narrative} What is the cutoff for the top 3% of the salaries? ANS: $55,960 PTS: 1 REF: SECTION 8.2 115. A worker earns $16 per hour at a plant and is told that only 5% of all workers make a higher wage. If the wage is assumed to be normally distributed and the standard deviation of wage rates is $5 per hour, find the average wage for the plant workers per hour. ANS: P(X > 16) = .05 (16 ) / 5 = 1.645 = $7.78 PTS: 1 REF: SECTION 8.2 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.