Section 7.2 Applications of the Normal Distribution 309 39. Tire lifetimes: The lifetime of a certain type of automobile tire (in thousands of miles) is normally distributed with mean �� = 40 and standard deviation �� = 5. a. What is the probability that a randomly chosen tire has a lifetime greater than 48 thousand miles? 0.0548 b. What proportion of tires have lifetimes between 38 and 43 thousand miles? 0.3811 Tech: 0.3812 c. What proportion of tires have lifetimes less than 46 thousand miles? 0.8849 40. Tree heights: Cherry trees in a certain orchard have heights that are normally distributed with mean �� = 112 inches and standard deviation �� = 14 inches. a. What proportion of trees are more than 120 inches tall? 0.2843 Tech: 0.2839 b. What proportion of trees are less than 100 inches tall? 0.1949 Tech: 0.1957 c. What is the probability that a randomly chosen tree is between 90 and 100 inches tall? 0.1367 Tech: 0.1376 41. Tire lifetimes: The lifetime of a certain type of automobile tire (in thousands of miles) is normally distributed with mean �� = 40 and standard deviation �� = 5. a. Find the 15th percentile of the tire lifetimes. 34.80 Tech: 34.82 b. Find the 68th percentile of the tire lifetimes. 42.35 Tech: 42.34 c. Find the first quartile of the tire lifetimes. 36.65 Tech: 36.63 d. The tire company wants to guarantee that its tires will last at least a certain number of miles. What number of miles (in thousands) should the company guarantee so that only 2% of the tires violate the guarantee? 29.75 Tech: 29.73 42. Tree heights: Cherry trees in a certain orchard have heights that are normally distributed with mean �� = 112 inches and standard deviation �� = 14 inches. a. Find the 27th percentile of the tree heights. 103.46 Tech: 103.42 b. Find the 85th percentile of the tree heights. 126.56 Tech: 126.51 c. Find the third quartile of the tree heights. 121.38 Tech: 121.44 d. An agricultural scientist wants to study the tallest 1% of the trees to determine whether they have a certain gene that allows them to grow taller. To do this, she needs to study all the trees above a certain height. What height is this? 43. How much is in that can? The volume of beverage in a 12-ounce can is normally distributed with mean 12.05 ounces and standard deviation 0.02 ounce. a. What is the probability that a randomly selected can will contain more than 12.06 ounces? 0.3085 b. What is the probability that a randomly selected can will contain between 12 and 12.03 ounces? 0.1525 Tech: 0.1524 c. Is it unusual for a can to be underfilled (contain less than 12 ounces)? Yes 44. How much do you study? A survey among freshmen at a certain university revealed that the number of hours spent studying the week before final exams was normally distributed with mean 25 and standard deviation 7. a. What proportion of students studied more than 40 hours? b. What is the probability that a randomly selected student spent between 15 and 30 hours studying? 0.6847 Tech: 0.6859 c. What proportion of students studied less than 30 hours? 45. How much is in that can? The volume of beverage in a 12-ounce can is normally distributed with mean 12.05 ounces and standard deviation 0.02 ounce. a. Find the 60th percentile of the volumes. 12.055 b. Find the 4th percentile of the volumes. 12.015 c. Between what two values are the middle 95% of the volumes? 12.011, 12.089 46. How much do you study? A survey among freshmen at a certain university revealed that the number of hours spent studying the week before final exams was normally distributed with mean 25 and standard deviation 7. a. Find the 98th percentile of the number of hours studying. b. Find the 32nd percentile of the number of hours studying. c. Between what two values are the middle 80% of the hours spent studying? 16.04 and 33.96 Tech: 16.03 and 33.97 47. Precision manufacturing: A process manufactures ball bearings with diameters that are normally distributed with mean 25.1 millimeters and standard deviation 0.08 millimeter. a. What proportion of the diameters are less than 25.0 millimeters? 0.1056 b. What proportion of the diameters are greater than 25.4 millimeters? 0.0001 c. To meet a certain specification, a ball bearing must have a diameter between 25.0 and 25.3 millimeters. What proportion of the ball bearings meet the specification? 0.8882 Tech: 0.8881 48. Exam grades: Scores on a statistics final in a large class were normally distributed with a mean of 75 and a standard deviation of 8. a. What proportion of the scores were above 90? b. What proportion of the scores were below 65? 0.1056 c. What is the probability that a randomly chosen score is between 70 and 80? 0.4714 Tech: 0.4680 49. Precision manufacturing: A process manufactures ball bearings with diameters that are normally distributed with mean 25.1 millimeters and standard deviation 0.08 millimeter. a. Find the 60th percentile of the diameters. b. Find the 32nd percentile of the diameters. c. A hole is to be designed so that 1% of the ball bearings will fit through it. The bearings that fit through the hole will be melted down and remade. What should the diameter of the hole be? 24.9136 Tech: 24.9138 d. Between what two values are the middle 50% of the diameters? 25.0464 and 25.1536 Tech: 25.0460 and 25.1540 50. Exam grades: Scores on a statistics final in a large class were normally distributed with a mean of 75 and a standard deviation of 8. a. Find the 40th percentile of the scores. 73.00 Tech: 72.97 b. Find the 65th percentile of the scores. 78.12 Tech: 78.08 c. The instructor wants to give an A to the students whose scores were in the top 10% of the class. What is the minimum score needed to get an A? 85.24 Tech: 85.25
navidi_monk_elementary_statistics_2e_ch7-9
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