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Part III: Sample Chapter Tests and Answers CHAPTER 5 TEST 95 FORM A PAGE 1 1. Sam is a representative who sells large appliances such as refrigerators, stoves, and so forth. Let x = number of appliances Sam sells on a given day. Let f = frequency with which he sells x appliances. For a random sample of 240 days, Sam had the following sales record. x 0 1 2 3 4 5 6 7 f 9 72 63 41 28 14 8 5 Assume that the sales record is representative of the population of all sales days. (a) Use the relative frequency to find P(x) for x = 0 to 7. (b) Use a histogram to graph the probability distribution of part (a). 1. (a) __________________________ (b) (c) Compute the probability that x is between 2 and 5 (including 2 and 5). (c) __________________________ (d) Compute the probability that x is less than 3. (d) __________________________ (e) Compute the expected value of the x distribution. (e) __________________________ (f) Compute the standard deviation of the x distribution. (f)_ __________________________ 2. The director of a health club conducted a survey and found that 23% of members used only the pool for workouts. Based on this information, what is the probability that, for a random sample of 10 members, 4 used only the pool for workouts? 2. ______________________________ 3. Of those mountain climbers who attempt Mt. McKinley, only 65% reach the summit. In a random sample of 16 mountain climbers who are going to attempt Mt. McKinley, what is the probability of each of the following? (a) All 16 reach the summit. 3. (a) __________________________ (b) At least 10 reach the summit. (b) __________________________ (c) No more than 12 reach the summit. (c) __________________________ (d) From 9 to 12 reach the summit, including 9 and 12. (d) __________________________ Copyright © Houghton Mifflin Company. All rights reserved. Instructor's Resource Guide Understandable Statistics, 9/e 96 CHAPTER 5 TEST FORM A PAGE 2 4. A coach found that about 12% of all hockey games end in overtime. What is the expected number of games ending in overtime if 50 hockey games are played as a random sample? 4. _____________________________ 5. There is about 75% probability that a truck will be going over the speed limit on I-80 between Cheyenne and Rock Springs, Wyoming. Suppose that a random sample of five trucks on this stretch of I-80 is observed. (a) Make a histogram showing the probability that r = 0, 1, 2, 3, 4, 5 trucks are speeding. 5. (a) (b) Find the mean of this probability distribution. (b) __________________________ (c) Find the standard deviation of this probability distribution. (c) __________________________ 6. Records show that the probability of catching a northern pike over 40 inches at Taltson Lake (Canada) is about 15% for each full day a person spends fishing. What is the minimum number of days a person must fish to be at least 83.3% sure of catching one or more northern pike over 40 inches? 6. _____________________________ 7. We are interested in when the first six will occur for repeated rolls of a balanced die. What is the population mean for this geometric distribution (i.e., the expected number of rolls for the first 6 to occur)? 7. _____________________________ Copyright © Houghton Mifflin Company. All rights reserved. Part III: Sample Chapter Tests and Answers CHAPTER 5 TEST 97 FORM A PAGE 3 8. The probability that an airplane is more than 45 minutes late on arrival is about 15%. Let n = 1, 2, 3, … represent the number of times a person travels on an airplane until the first time the plane is more than 45 minutes late. (a) Write a brief but complete discussion in which you explain why the geometric distribution would be appropriate. Write out a formula for the probability distribution of the random variable n. 8. (a) __________________________ (b) What is the probability that the third time a person flies is the first time the person is late by more the 45 minutes? (b) __________________________ (c) What is the probability that more than three flights are required before a plane is more than 45 minutes late? (c) __________________________ 9. Suppose that the average number of customers entering a store in a 20-minute period is six customers. The store wants a probability distribution for the number of people entering the store each 20 minutes. (a) Write a brief but complete discussion in which you explain why the Poisson approximation to the binomial would be appropriate. Are the assumptions satisfied? What is ? Write out a formula for the probability distribution of r. (b) What is the probability that exactly three customers enter the store during a 20-minute period? 9. (a) __________________________ (b) __________________________ (c) What is the probability that more than three customers enter the store during a 20-minute period? (c) __________________________ 10. The probability that a new medication will cause a bad side effect is 0.03. The new medication has been given to 150 volunteers. Let r be the random variable representing the number of people who have a bad side effect. (a) Write a brief but complete discussion in which you explain why the Poisson approximation to the binomial would be appropriate. Are the assumptions satisfied? What is ? Write out a formula for the probability distribution of r. 10. (a) __________________________ (b) Compute the probability that exactly 3 people from the sample of 150 volunteers will have a bad side effect from the medication. (b) __________________________ (c) Compute the probability that more than 3 people from the sample of 150 volunteers will have a bad side effect from the medication. (c) __________________________ Copyright © Houghton Mifflin Company. All rights reserved. Instructor's Resource Guide Understandable Statistics, 9/e 98 CHAPTER 5 TEST FORM B PAGE 1 1. An aptitude test was given to a random sample of 228 people intending to become data-entry clerks. The results are shown below, where x is the score on a 10-point scale and f is the frequency of people with this score. x 1 2 3 4 5 6 7 8 9 10 f 9 21 46 51 42 18 12 10 8 5 Assume that these data represent the entire population of people intending to become data-entry clerks. (a) Use the relative frequencies to find P(x) for x = 1 to x = 10. (b) Use a histogram to graph the probability distribution of part (a). 1. (a) __________________________ (b) (c) To be accepted into a training program, students must have a score of 4 or higher. What is the probability that an applicant selected at random will have this score? (c) __________________________ (d) To receive a tuition scholarship, a student needs a score of 8 or higher. What is the probability that an applicant selected at random will have such a score? (d) __________________________ (e) Compute the expected value of the x distribution. (e) __________________________ (f) Compute the standard deviation of the x distribution. (f) __________________________ 2. The management of a restaurant conducted a survey and found that 28% of the customers preferred to sit in the smoking section. Based on this information, what is the probability that for a random sample of 12 customers, 3 preferred the smoking section? 2. _____________________________ 3. Of all college freshmen who try out for the track team, the coach will only accept 30%. If 15 freshmen try out for the track team, what is the probability that (a) all 15 are accepted? 3. (a) __________________________ (b) at least 8 are accepted? (b) __________________________ (c) no more than 4 are accepted? (c) __________________________ (d) between 5 and 10 are accepted (including 5 and 10)? (d) __________________________ Copyright © Houghton Mifflin Company. All rights reserved. Part III: Sample Chapter Tests and Answers CHAPTER 5 TEST 99 FORM B 4. The president of a bank approves 68% of all new applications. What is the expected number of approvals if 75 loan applications are chosen at random? PAGE 2 4. _____________________________ 5. The probability that a vehicle will change lanes while making a turn is 55%. Suppose that a random sample of seven vehicles is observed making turns at a busy intersection. (a) Make a histogram showing the probability that r = 0, 1, 2, 3, 4, 5, 6, 7 vehicles will make a lane change while turning. 5. (a) (b) Find the expected value of this probability distribution. (b) __________________________ (c) Find the standard deviation of this probability distribution. (c) __________________________ 6. Past records show that the probability of catching a lake trout over 15 pounds at Talston Lake (Canada) is about 20% for each full day a person spends fishing. What is the minimal number of days a person must fish to be at least 89.3% sure of catching one or more lake trout over 15 pounds? 6. _____________________________ 7. We are interested in when the first odd number will occur for repeated rolls of a balanced die. What is the population mean for this geometric distribution (i.e., the expected number of rolls for the first odd number to occur)? 7. _____________________________ Copyright © Houghton Mifflin Company. All rights reserved. Instructor's Resource Guide Understandable Statistics, 9/e 100 CHAPTER 5 TEST FORM B PAGE 3 8. Past records at an appliance store show that about 60% of the customers who look at appliances will buy. Let n = 1, 2, 3, … represent the number of customers a sales clerk must help until the first sale of the day. (a) Write a brief but complete discussion in which you explain why the geometric distribution would apply in this context. Write out a formula for the probability distribution of the random variable n. 8. (a) __________________________ (b) Compute P(n = 4). (b) __________________________ (c) Compute P(n 3). (c) __________________________ 9. At Community Hospital maternity ward, babies arrive at an average of eight babies per day. The hospital staff wants a probability distribution for the number of babies arriving each day. (a) Write a brief but complete discussion in which you explain why the Poisson distribution would be appropriate. What is ? Write out a formula for the probability distribution. 9. (a) __________________________ (b) What is the probability that exactly seven babies are born during the next day? (b) __________________________ (c) What is the probability that fewer than three babies are born during the next day? (c) __________________________ 10. As a telecommunications satellite goes over the horizon, stored messages are relayed to the next satellite, which is still in position. However, the probability is 0.01 that an interruption will occur, and the relay transmission will be lost. Of 200 such relays, let r be the random variable that represents the number of transmissions that are lost. (a) Write a brief but complete discussion in which you explain why the Poisson approximation to the binomial would be appropriate. Are the assumptions satisfied? What is ? Write out a formula for the probability distribution of r. 10. (a) __________________________ (b) Compute the probability that exactly two transmissions are lost. (b) __________________________ (c) Compute the probability that more than two transmissions are lost. (c) __________________________ Copyright © Houghton Mifflin Company. All rights reserved. Part III: Sample Chapter Tests and Answers CHAPTER 5 TEST 101 FORM C PAGE 1 Write the letter of the response that best answers each problem. 1. The following data are based on a survey taken by a consumer research firm. In this table, x = number of televisions in household and % = percentages of U.S. households. x 0 1 2 3 4 5 or more % 3% 11% 28% 39% 12% 7% A. What is the probability that a household selected at random has less than three televisions? (a) 0.81 (b) 0.39 (c) 0.42 (d) 0.58 (e) 0.19 B. What is the probability that a household selected at random has more than four televisions? (a) 0.7 (b) 0.19 (c) 0.81 (d) 0.93 (b) 2.67 (c) 1.28 (d) 1.13 2. (c) 1.28 (d) 1.13 (b) 0.67 (c) 0.28 (d) 0.13 D. __________ (e) 3.1 A meteorologist found from the past year’s records that it rained 17% of the days. Based on this information, what is the probability that, for a random sample of 15 days, it rained 3 of those days? (a) 0.17 3. (b) 2.67 C. __________ (e) 3.1 D. Compute the standard deviation of the x distribution (round televisions of five or more to five). (a) 15 B. __________ (e) 0.07 C. Compute the expected value of the x distribution (round televisions of five or more to five). (a) 15 1. A. __________ 2. _____________ (e) 0.24 Of those people who lose weight on a diet, 90% gain all the weight back. In a random sample of 12 dieters who have lost weight, what is the probability of each of the following? A. All 12 gain the weight back. (a) 0.90 (b) 0.282 3. A. __________ (c) 0.540 (d) 0.142 (e) 10.8% B. At least 9 gain the weight back. (a) 0.974 (b) 0.026 (c) 0.889 B. __________ (d) 1.33% (e) 0.997 C. No more than 6 gain the weight back. (a) 0.004 (b) 1.000 (c) 0.999 C. __________ (d) 0.000 (e) 0.531 D. From 8 to 10 gain the weight back, including 8 and 10. (a) 0.085 (b) 1.17% (c) 0.336 Copyright © Houghton Mifflin Company. All rights reserved. (d) 0.387 D. __________ (e) 0.118 Instructor's Resource Guide Understandable Statistics, 9/e 102 CHAPTER 5 TEST FORM C PAGE 2 4. The manager of a supermarket found that 72% of the shoppers who taste a free sample of a food item will buy the item. What is the expected number of shoppers that will buy the item if 50 randomly selected shoppers taste a free sample? (a) 10 (b) 36 (c) 3 (d) 72 4. _____________ (e) 50 5. The probability that merchandise stolen from a store will be recovered is 15%. Suppose that a random sample of eight stores, from which merchandise has been stolen, is chosen. A. Find the mean of this probability distribution. (a) 1.02 (b) 1.07 (c) 1.01 5. A. __________ (d) 1.14 (e) 1.2 B. Find the standard deviation of this probability distribution. (a) 1.02 (b) 1.07 (c) 1.01 (d) 1.14 B. __________ (e) 1.2 6. Records show that the probability of seeing a hawk migrating on a day in September is about 35%. What is the minimum number of days a person must watch to be at least 96.8% sure of seeing one or more hawks migrating? (a) 5 (b) 6 (c) 7 (d) 8 (e) 9 7. We are interested in when the first 6 will occur for repeated rolls of a balanced die. What is the population mean for this geometric distribution (i.e., the expected number of rolls for the first 6 to occur)? (a) 6 (b) 1 6 (c) 7 6. _____________ (d) 8 7. _____________ (e) 9 8. Rita is studying to be a real estate agent. About 61% of all people who take the licensing exam pass. Let n = 1, 2, 3, … represent the number of times a person takes the exam until he or she passes. A. What is the formula for the probability distribution of the random variable n. (a) P(n) = (0.61)n(0.39)n1 (b) P(n) = 0.39(0.61)n1 (c) P(n) = 0.61(0.39)n1 (d) P(n) = (0.61)n1(0.39)n (e) P(n) = (0.39)n B. What is the probability that Rita will need three attempts to pass the exam? (a) 0.227 (b) 0.036 (c) 0.145 (d) 0.093 (b) 0.059 (c) 0.152 (d) 0.023 B. __________ (e) 0.059 C. What is the probability that Rita will need more than three attempts to pass the exam? (a) 0.941 8. A. __________ C. __________ (e) 0.907 Copyright © Houghton Mifflin Company. All rights reserved. Part III: Sample Chapter Tests and Answers CHAPTER 5 TEST 103 FORM C PAGE 3 9. Suppose that the average number of customers calling a technical support number in a 10-minute period is seven customers. The company wants a probability distribution for the number of people calling each 10 minutes. A. What is the formula for the Poisson probability distribution? 7 r (a) P r e 7 7 (b) P r e rr ! 7 7 (c) P r e rr ! 7 r (d) P r 7 r ! 7 r! 9. A. __________ 7 r (e) P r e 7 r! e B. What is the probability that exactly four customers call the support number during a 10-minute period? (a) 0.0912 (b) 0.9088 (c) 0.0521 (d) 0.1729 (e) 0.5714 C. What is the probability that more than four customers call the support number during a 10-minute period? (a) 0.1729 (b) 0.8271 (c) 0.0817 (d) 0.9183 B. __________ C. __________ (e) 0.9088 10. The probability that a manufactured part at a plant is defective is 0.02. The plant has manufactured 300 parts. Let r be the random variable representing the number of defective parts. A. What is the formula for the Poisson approximation to the binomial probability distribution of r? (a) P r e0.02 300 r! r 0.02 0.02r (c) P r e r! 10. A. __________ 6 r (b) P r e 6 r! 6 r (d) P r e 6 r! 6 (e) P r e rr ! 6 B. What is the probability that exactly five parts are defective? (a) 0.0268 (b) 0.9732 (c) 0.8394 (d) 0.0000 B. __________ (e) 0.1606 C. What is the probability that fewer than two parts are defective? (a) 0.0620 (b) 0.9380 (c) 0.9826 Copyright © Houghton Mifflin Company. All rights reserved. (d) 0.0174 C. __________ (e) 0.9999