Chapter 6 Normal Distributions

Chapter 6 Normal Distributions (S) 1. Statistical Abstracts (117th edition) provides information about state per capita income taxes. The distribution is mound shaped with mean µ = $1,701 and standard deviation σ = $672. (a) Use the empirical rule to find an interval centered about the mean in which about 68% of the data will fall. (b) Estimate a range of values centered about the mean in which about 95% of the state per capita income taxes will fall. (M) 2. Which statement is not an important property of a normal curve? A. The curve is bell-shaped with the highest point over the mean µ. B. The transition points between cupping upward and downward occur above µ + σ and µ σ. C. Approximately 99.8% of the data values will be within four standard deviations on each side of the mean. D. It is symmetrical about a vertical line through µ. E. The curve approaches the horizontal axis but never touches or crosses it. (S) (S) 3. The running times of feature movies have a mound shaped distribution with mean µ = 110 minutes and standard deviation σ = 30 minutes. (a) Use the empirical rule to find an interval centered about the mean in which 68% of the data will fall. (b) Estimate a range of values centered about the mean in which about 95% of the data will fall. 4. The percentage growth rate of stocks of companies that specialize in construction and residential home building has a mound shaped distribution with mean approximately 9.8% and standard deviation 3.8%. (a) Estimate a range of values centered about the mean in which about 68% of the data will fall. (b) Estimate a range of values centered about the mean in which about 95% of the data will fall. (M) 5. In a normal distribution, if µ = 6 and σ = 3, where are the transition points located? A. 0, 12 B. 3, 15 C. 4, 8 D. 3, 9 E. 5, 7 163

164 Test Item File Understandable Statistics, 7th Edition (S) (S) 6. The percentage growth rate of stocks of companies that specialize in recreation equipment and appliances has a mound shaped distribution with mean estimated to be 13.1% and standard deviation 6.0%. (a) Estimate a range of values centered about the mean in which 68% of the data will fall. (b) Estimate a range of values centered about the mean in which 95% of the data will fall. 7. A certain brand of light bulb lasts according to a normal distribution with µ = 1000 hr and σ = 50 hr. (a) What is the probability that a light bulb selected at random will last between 1000 and 1050 hr? (b) What is the probability that a light bulb selected at random will last between 850 hr and 900 hr? (S) 8. Eggs Inc. is a restaurant that specializes in breakfasts. The company has established a target mean of 175 customers per morning with standard deviation 25. For 12 consecutive days the number of breakfasts was tabulated. In the table below t = time in days and x = number of breakfasts. t 1 2 3 4 5 6 7 8 9 10 11 12 x 165 115 110 130 180 192 241 199 183 212 188 235 (S) (a) Make a control chart for the above data. (b) Determine whether the process is in statistical control. If it is not, specify which out-of-control signals are present. 9. The Highway Patrol has a target of 28 traffic tickets per week with standard deviation 5 tickets per week for a stretch of mountain highway. The number of tickets issued for 15 consecutive weeks is given below. t = number of the week, x = number of tickets. t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x 22 20 15 16 25 30 30 34 30 32 36 40 33 35 30 (a) Make a control chart for the above data. (b) Determine whether the process is in statistical control. If it is not specify which out-of-control signals are present. (M) 10. The lifetime of a disposable water filter is normally distributed with mean µ = 30 gal and standard deviation σ = 6 gal. What is the probability that a filter selected at random will last from 36 gal to 42 gal? A. 0.27 B. 0.135 C. 0.34 D. 0.475 E. 0.1585

Chapter 6 Normal Distributions 165 (S) 11. Lewis earned 85 on his biology midterm and 81 on his history midterm. In the biology class the mean score was 79 with standard deviation 5. In the history class the mean score was 76 with standard deviation 3. (a) Convert each score to a standard z score. (b) On which test did he do better compared to the rest of the class? (S) 12. Last fiscal year, the average number of defective televisions produced by a factory each day was 17.6, with a standard deviation of 5.2. For the first ten days of this fiscal year, the number of defective televisions produced is shown in the following table: Day 1 2 3 4 5 6 7 8 9 10 Defective Televisions 19 15 13 16 24 29 34 25 20 10 (a) Make a control chart for the daily number of defective televisions produced. (b) Do the data indicate that the production is in control? Explain your answer. (S) 13. Statistical Abstracts (117th edition) lists the number of federal officials indicted for the years 1985 to 1995. The mean number of officials indicted in this period is 574 and the standard deviation is 184. In the table below t = number of year (with year 1 = 1985) and x = number of indictments. t 1 2 3 4 5 6 7 8 9 10 11 x 563 596 651 629 695 615 803 624 627 571 527 (a) Make a control chart for the above data. (b) Determine if the process is in statistical control. If it is not, specify which outof-control signals are present. (S) 14. Jan earned 86 on her political science midterm and 82 on her chemistry midterm. In the political science class the mean score was 80 with standard deviation 4. In the chemistry class the mean score was 70 with standard deviation 6. (a) Convert each midterm score to a standard z score. (b) On which test did she do better compared to the rest of the class? (M) 15. A control chart contains the data points: 11 15 16 17 16 18 20 19 16 17 13 10 12 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 µ = 14.3 and σ = 3.1. Which out-of-control indicator is evident from this data? A. At least four points beyond the σ line. B. At least two of three consecutive points lie beyond the 2σ level on the same side of the center line C. One point falls beyond the 3σ level D. One point falls beyond the 2σ level E. A run of nine consecutive points on one side of the center line

166 Test Item File Understandable Statistics, 7th Edition (S) 16. Bob earned 116 on his accounting midterm and 82 on his biology midterm. For the accounting exam the mean score was 102 with standard deviation 14. For the biology exam the mean score was 70 with standard deviation 5. (a) Convert each midterm score to a standard score. (b) On which test did he do better compared to the rest of the class? (S) 17. Troy took a standardized test to try to get credit for first-year Spanish by examination. His standardized score was 1.9. The mean score for the exam was 100 with standard deviation 12. The language department requires a raw score of 120 to get credit by exam for first-year Spanish. (a) Compute Troy s raw score. (b) Will Troy get credit for first-year Spanish based on this exam? (M) 18. The graduation rate for seniors at a certain university in any particular year is, on average, 72.1% and is normally distributed. The population standard deviation is 3.5%. What is the probability, in any given year, that between 79.1% and 82.6% of the seniors will graduate? A. 0.0235 B. 0.047 C. 0.135 D. 0.27 E. 0.4985 (S) 19. Leo has a pizza business. The times to prepare and deliver pizzas are approximately normally distributed with mean µ = 20 minutes and standard deviation σ = 10 minutes. (a) Suppose that Leo advertises If it takes more than 30 minutes to get your pizza, you get it free. What fraction of his take-out pizzas will he have to give away free? (b) How long a time should Leo allow if he wants to give away no more than 5% of the pizzas? (S) 20. Weights of Pacific yellowfin tuna follow a normal distribution with mean weight 68 lb and standard deviation 12 lb. A Pacific yellowfin tuna is caught at random. (a) Find the probability that its weight is less than 48 lb. (b) Find the probability that its weight is more than 100 lb. (c) Find the probability that its weight is between 48 lb and 100 lb. (M) 21. A normal distribution has µ = 5 and σ = 1.5. Find z for x = 2.5. A. z = 1.4 B. z = 1.66 C. z = 1.66 D. z = 1.67 E. z = 1.67 (S) 22. In order to make a little money for the holidays, Marla is raising turkeys. The male of the species she has selected has a mean weight of 17 lb at maturity and a standard deviation of 1.8 lb. The weights are known to be normally distributed. What is the probability of raising a male turkey that weighs at least 21 lb at maturity?

Chapter 6 Normal Distributions 167 (S) 23. The Snack Pack of potato chips is advertised to weigh 3.5 oz. The weights are normally distributed with mean 3.5 oz and standard deviation 0.2 oz. A Snack Pack of potato chips is selected at random. (a) Find the probability that it weighs less than 3.0 oz. (b) Find the probability that it weighs more than 3.7 oz. (M) 24. Jose received an 82 on his math exam. µ = 73 and σ = 8 for the entire class. Compute the number of standard deviations Jose s score is from the mean. A. z = 1.014 B. z = 1.125 C. z = 1.125 D. z = 1.014 E. z = 0.805 (S) 25. The snow pack on the summit of Wolf Creek Pass, Colorado on March 1 has been measured for many years. It is normally distributed with mean µ = 78.1 inches and standard deviation σ = 10.4 inches. A year is selected at random. (a) Find the probability that the snowpack is less than 60 inches. (b) Find the probability that the snowpack is more than 85 inches. (c) Find the probability that the snowpack is between 60 inches and 85 inches. (S) 26. In Manoa Valley on the island of Oahu (Hawaii) the annual rainfall averages 43.6 inches with standard deviation 7.5 inches. A year is chosen at random. (a) Find the probability that the annual rainfall is more than 56 inches. (b) Find the probability that the annual rainfall is less than 32 inches. (c) Find the probability that the annual rainfall is between 32 inches and 56 inches. (S) 27. A normal distribution has µ = 13 and σ = 2.1. To transform this into standard normal distribution, (a) how many units, and in which direction along the x-axis, must the distribution be shifted? (b) will the standard normal distribution be wider or narrower than the original? (S) 28. Find the z value so that (a) 8% of the area under the standard normal curve lies to the left of z. (b) 93% of the area under the standard normal curve lies between z and z. (S) 29. The life of the Corn Delight popcorn maker is normally distributed with mean 20 months and standard deviation 2 months. The manufacturer will replace a Corn Delight popper if it breaks during the guarantee period. (a) If the manufacturer guarantees the popper for 18 months what fraction of the poppers will he probably have to replace? (b) How long should the guarantee period be if the manufacturer does not want to replace more than 3% of the poppers? (Round to the nearest month.)

168 Test Item File Understandable Statistics, 7th Edition (M) 30. A normal distribution was transformed to a standard normal distribution. The original µ and σ were 17 and 3.4, respectively. For z = 2, what is the corresponding x value? A. 11.9 B. 37.4 C. 18.7 D. 7 E. 23.8 (S) 31. The operating life of a Sports Master fishing reel is normally distributed with mean 48 months and standard deviation 6 months. The manufacturer will replace any Sports Master reel that malfunctions during the guarantee period. (a) If the guarantee period is 45 months, what fraction of the reels will Sports Master have to replace? (b) How long should the guarantee period be if the manufacturer does not want to replace more than 5% of the reels? (M) 32. Use a standard normal distribution table to find the area under the curve to the left of z = 1.17. A. 0.1190 B. 0.1210 C. 0.1423 D. 0.8790 E. 0.8577 (S) 33. The operating life of an Arctic Air portable cooler is normally distributed with mean 75 months and standard deviation 9 months. The manufacturer will replace any cooler that breaks during the guarantee period. (a) If the guarantee period is for 64 months, what fraction of the coolers will the manufacturer have to replace? (b) How long should the guarantee run if the manufacturer does not want to replace more than 2.5% of the coolers? (S) 34. Recently an investor tabulated the percent growth for a random sample of 25 large mutual funds. The distribution was mound shaped and symmetric. It had a mean µ = 25.7% and standard deviation σ = 7.1%. (a) Estimate an interval about the mean in which about 68% of these percent gains would fall. (b) Estimate an interval centered about the mean in which about 95% of these percent gains would fall. (M) 35. Use a standard normal distribution table to find the area under the curve to the left of z = 1.57. A. 0.0582 B. 0.9292 C. 0.9418 D. 0.9525 E. 0.0708

Chapter 6 Normal Distributions 169 (S) 36. George is a senior citizen who takes medication to stabilize his (systolic) blood pressure. He takes his blood pressure every day and, from long experience, he knows that the mean pressure is 137 with standard deviation 6. Assume the blood pressure follows a normal distribution. For the past ten days his blood pressure has been recorded in the following table. t = number of the day. x = blood pressure. t 1 2 3 4 5 6 7 8 9 10 x 128 133 130 134 139 140 139 142 158 161 (a) Make a control chart for the systolic blood pressure readings. (b) Determine whether the blood pressure distribution is in statistical control. If it is not, indicate which out-of-control signals are present. (c) If George s blood pressure changes too much it is recommended that his medication be changed. Looking at the control chart, do you think it might be time to consider changing his medication? Explain. (S) 37. Let x represent the life of a AA battery in a portable radio. The x distribution has µ = 44 hours and σ = 3.6 hours. Convert each of the following x intervals into z intervals. Round to the nearest hundredth if necessary. (a) 40 x 46 (b) 37 x 51 (c) 41 x 45 (S) 38. From long experience the store manager at the local W-Mart Store knows that the number of customers in the store each day follows a normal distribution with mean 611 customers and standard deviation 36 customers. The table below gives the number of customers in the store each day for the last ten days. t = number of the day, x = number of customers. t 1 2 3 4 5 6 7 8 9 10 x 541 592 620 605 548 535 510 489 500 490 (a) Make a control chart for the daily number of customers at W-Mart. (b) Determine if the distribution of customers is in statistical control. If it is not, indicate which out-of-control signals are present. (c) The manager of W-Mart usually does more advertising and has more sales if the number of customers drops off. Looking at the control chart what would you recommend at this time? (M) 39. Use a standard normal distribution table to find the area under the curve between z = 1.08 and z = 2.13. A. 0.8433 B. 0.1567 C. 0.8234 D. 0.1766 E. 0.8339

170 Test Item File Understandable Statistics, 7th Edition (S) 40. Let x represent the life of a 60-watt light bulb. The x distribution has a mean µ = 1,000 hours with standard deviation σ = 75 hours. Convert each of the following z intervals into x intervals. (a) 0 z 1.25 (b) 1.5 z 2.4 (c) 1.25 z 2.25 (S) 41. A high school counselor was given the following z intervals in information about a vocational training aptitude test. The test scores had a mean µ = 450 points and standard deviation σ = 35 points. Convert each interval into a x interval. x = test score. (a) 1.14 z 2.27 (b) z 2.88 (c) z 1.85 (M) 42. The following are six grade-point averages for six students. Joseph: 2.01 Sabine: 2.64 Xena: 3.67 Ron: 2.38 Maria: 3.22 Ali: 3.59 If these grade-point averages were used to create a standard normal distribution, which student s grade-point average would have a z-value of 0.488? A. Xena B. Maria C. Sabine D. Ron E. Ali (S) 43. David and Laura are both applying for a position on the ski patrol. David took the advanced first aid class at his college. His score on the comprehensive final exam was 173 points. The final exam scores followed a normal distribution with mean 150 points and standard deviation 25 points. Laura took an advanced first aid class at the plant where she works. For the method of testing there her cumulative score on all exams was 88. The cumulative scores followed a normal distribution with mean 65 and standard deviation 10. Both courses are comparable in content and level of difficulty. There is only one position available on the ski patrol and David and Laura are equally qualified as skiers. (a) Both David and Laura scored 23 points above the mean for their respective tests. Does this mean that they both gave the same performance on the first aid course? Explain your answer. (b) Calculate the z scores for David and Laura. On the basis of test results which of them should get the job? Explain your answer.

Chapter 6 Normal Distributions 171 (S) 44. Jim scored 630 on at national bankers examination in which the mean is 600 and the standard deviation is 70. June scored 530 on the Hoople College bankers examination for which the mean is 500 and the standard deviation is 25. (a) Calculate the standard z score for Jim and for June. (b) If Jim and June both apply for a job at Hoople State Bank and each examination has equal weight in the hiring criteria, who has the better chance based on test results? Explain your answer. (M) 45. Let x have a normal distribution with µ = 13 and σ = 3. Find the probability that an x value selected at random from this distribution is between 8 and 16. A. 0.8413 B. 0.0062 C. 0.9938 D. 0.1649 E. 0.7938 (S) 46. Researchers at a pharmaceutical company have found that the effective time duration of a safe dosage of a pain relief drug is normally distributed with mean 2 hours and standard deviation 0.3 hour. For a patient selected at random: (a) What is the probability that the drug will be effective for 2 hours or less? (b) What is the probability that the drug will be effective for 1 hour or less? (c) What is the probability that the drug will be effective for 3 hours or more? (S) 47. Quality control for Comfort-Ease Computer Peripherals, Inc. has done studies showing that its voice recognition device has a mean life of 7.5 years with standard deviation 2.2 years. The manufacturer will replace any such device that wears out during the guarantee period. (a) If the guarantee period is 5 years, what fraction of the voice recognition devices will the manufacturer have to replace? (b) How long should the guarantee run if the manufacturer wishes to replace no more than 10% of the devices? (M) 48. Let x have a normal distribution with µ = 20 and σ = 4. Find the probability that an x value selected at random from this distribution is between 19 and 21. A. 0.8026 B. 0.5987 C. 0.1974 D. 0.4013 E. 1.0000 (S) 49. The life of a Freeze Breeze electric fan is normally distributed with mean 4 years and standard deviation 1.2 years. The manufacturer will replace any fan that wears out during the guarantee period. (a) What fraction of the fans will have to be replaced if the fans are guaranteed for three years? (b) How long should the fans be guaranteed if the manufacturer does not want to replace more than 5% of them? (Give the answer to the nearest month.)

172 Test Item File Understandable Statistics, 7th Edition (S) 50. The Zinger is a modern sports car with an epoxy body. The manufacturer constructs Zinger bodies under pressure in a mold, and must wait a certain length of time to be sure that enough resin bonding and hardening have occurred before removing the body from the mold. Chemical engineers have found that the hardening time in normally distributed with mean 28.7 hours and standard deviation 6.4 hours. What is the minimal length of time the body should stay in the mold if the manufacturer wants to be 85% sure that the bonds have hardened? (M) 51. Which of the following ranges of z values corresponds to a probability of 0.8213? A. 1.65 z 2 B. 1.13 z 1.65 C. 1.13 z 1.91 D. 1 z 1 E. 1 z 1.65 (S) 52. As the manager of a large manufacturing plant, you are concerned about employee absenteeism. The plant is staffed so that operations are still efficient when the average number of employees absent each shift has mean µ = 10.5 with standard deviation σ = 2. For the most recent 12 shifts the number of absent employees is given in the table below. t = number of shift, x = number absent. t 1 2 3 4 5 6 7 8 9 10 11 12 x 8 4 7 10 9 11 9 15 12 16 9 11 (a) Make a control chart for the above data. (b) Determine whether the number of workers absent is in statistical control. If it is not specify which out-of-control signals are present. (S) 53. A certain portable CD player is known to function properly for a mean 3.6 years with a standard deviation of 0.6 year. The player is sold with a 3-year warranty. What is the probability that the player will break during the warranty period? (S) 54. Peter is a door-to-door sales representative who makes a sale at only 8% of his house calls. If Peter makes 219 house calls today, what is the probability that: (a) he makes a sale at 12 or fewer houses? (b) he makes a sale at 20 or fewer houses? (c) he makes a sale at 15 to 25 houses? (S) 55. A toll-free computer software support line for the Magnet Computer System has established a target length of time for each customer call. The calls are targeted to have a mean duration of 7 minutes with standard deviation 2 minutes. For one help technician the most recent ten calls had duration given in the table below. t = number of call, x = duration. t 1 2 3 4 5 6 7 8 9 10 x 8 4 7 10 9 11 9 15 12 16 (a) Make a control chart showing the lengths of calls. (b) Determine whether the length of calls is in statistical control. If it is not specify which out-of-control signals are present.

Chapter 6 Normal Distributions 173 (S) 56. An air conditioner repairman is facing significant competition in his local area, so he wants to give a guarantee to his customers when he services their air conditioners. According to the data he has collected over the years, the mean time between necessary servicing of an air conditioner is 26 months, with a standard deviation of 6 months. How long should his guarantee period be (in months) if he does not want to perform a free service for more than 10% of his customers? (S) 57. In Summit County 35% of registered voters are Democrats. What is the probability that in a random sample of 300 voters: (a) 100 or more are Democrats? (b) between 100 and 120 are Democrats? (c) 90 or fewer are Democrats? (S) 58. Quality control reports that are not available to consumers indicate that 20% of Hot Shot ovens have faulty control panels. The owner of an apartment complex has just purchased 120 of these ovens. Assume that the ovens purchased constitute a random sample. (a) Can you use the normal approximation to the binomial distribution to estimate the probability that between 22 and 35 of them have faulty control panels? Explain. (b) Estimate the probability that between 22 and 35 of them (including 22 and 35) have faulty control panels. (M) 59. Find z such that 81% of the area under the standard normal curve lies between z and z. A. 1.29 B. 1.30 C. 1.31 D. 1.32 E. 1.33 (S) 60. Coast Guard inspectors have found that 16% of all lifeboats on large cruise ships have a faulty latch release. A large cruise ship has 73 lifeboats. What is the probability that 9 or more of the lifeboats have a faulty latch release? (M) 61. Find z such that 62% of the area under the standard normal curve lies between z and z. A. 0.19 B. 0.38 C. 0.86 D. 0.87 E. 0.88 (S) 62. Records at the College of Engineering show that 62% of all freshmen who declare EE (electrical engineering) as their intended major field of study eventually graduate with an EE major. This fall, the College of Engineering has 316 freshmen who have declared a EE major. (a) What is the probability that between 200 and 225 of them (including 200 and 225) will graduate with an EE major? (b) Can you use a normal approximation to the binomial distribution to estimate this probability? Explain.

174 Test Item File Understandable Statistics, 7th Edition (S) 63. One model of an imported automobile is known to have defective seat belts. An extensive study has found that 17% of the seat belts were incorrectly installed at the factory. A car dealer has just received a shipment of 196 cars of this model. What is the probability that in this sample: (a) 22 or fewer have defective seat belts? (b) 30 or more have defective seat belts? (c) between 25 and 50 have defective seat belts? (M) 64. Find z so that 3.5% of the standard normal curve lies to the right of z. A. 1.80 B. 1.81 C. 1.82 D. 0.04 E. 0.05 (S) 65. The probability that a person with a telephone has an unlisted number is 0.15. The district manager of a political action group is phoning people urging them to vote. In a district with 416 households, all households have phones. What is the probability that the district manager will find that (a) 50 or fewer households in the district have unlisted phone numbers? (b) 345 or more have listed phone numbers? (c) between 40 and 80 have unlisted numbers? (M) 66. Find z so that 42% of the standard normal curve lies to the left of z. A. 0.17 B. 0.18 C. 0.19 D. 0.20 E. 0.21 (S) 67. The probability of an adverse reaction to a flu shot is 0.02. If the shot is given to 1000 people selected at random, what is the probability that: (a) 15 or fewer people will have an adverse reaction? (b) 25 or more people will have an adverse reaction? (c) between 20 and 30 people will have an adverse reaction? (M) 68. For a binomial distribution with n = 100 and r = 83, using a normal probability distribution to approximate this binomial distribution, compute µ and σ. A. µ = 17; σ = 3.76 B. µ = 83; σ = 9.11 C. µ = 83; σ = 3.76 D. µ = 17; σ = 4.14 E. µ = 83; σ = 4.12 (S) 69. Canter Political Polling Service has found that the probability that a registered voter selected at random will answer and return a short questionnaire on a well-publicized issue is 0.31. If 1,000 questionnaires are sent to registered voters selected at random, what is the probability that: (a) 350 or more will be returned? (b) 320 or fewer will be returned? (c) between 315 and 355 will be returned?

Chapter 6 Normal Distributions 175 (M) 70. Bob received a z score of 0.9 on a college entrance exam. If the raw scores have a mean of 540 and a standard deviation of 80 points, what is his raw score? A. 612 B. 566 C 406 D. 468 E. 134 (S) 71. The state tourism board reports that 52% of the residents of the state plan to take summer vacations in the state. An independent company did a survey of 200 people. (a) Find the probability that fewer than 100 are planning in-state vacations. (b) Find the probability that 120 or more are planning in-state vacations. (M) 72. Determine P(z 2.09) with a standard normal distribution table. A. 0.0366 B. 0.9634 C. 0.9817 D. 0.183 E. 0.0183 (S) 73. A postal worker has observed that 72% of the customers who buy stamps request particular commemorative stamps. For a random sample of 80 customers find the probability that: (a) 50 or more ask for the special stamps. (b) From 50 to 65 people ask for the commemorative stamps. (Include 50 and 65.) (S) 74. On Professor Grindstone s final exam the mean score was 78 and the standard deviation 10. He wants to curve the exam so that the middle 50% of the students get C s. (a) Find a z score for which the area under the standard normal curve from z to z is 50%. (b) Find the range of scores which will be assigned a grade of C. (S) 75. The daily attendance of an economics class follows a normal distribution with a mean of 26 students and a standard deviation of four students. (a) Find the probability that the attendance will be less than 22 students. (b) Find the probability that the attendance will be 28 or more students. (S) 76. The daily attendance at a weekly flea market follows a normal distribution with mean 875 people and standard deviation 150 people. For a day chosen at random: (a) Find the probability that the attendance will be less than 500 people. (b) Find the probability that the attendance will be 700 people or more. (S) 77. The summer duck population at a lake in a state park is approximately normally distributed with a mean of 9640 ducks and a standard deviation of 820 ducks. Let x be the random variable that represents the size of the duck population in the summer of a given year. (a) Convert 7918 < x < 10,255 to a z interval. (b) Convert 1.25 < z < 0.90 to an x interval. (c) If 7590 represents an unusually low population, and 11,690 represents an unusually high number, give z numbers ranges that represent unusually low and high summer duck populations.

176 Test Item File Understandable Statistics, 7th Edition (S) 78. The attendance at home games of the college football team follows a normal distribution with mean 9,500 and standard deviation 600. For a game chosen at random: (a) Find the probability that the attendance will be between 8,000 and 11,000. (b) Find the probability that the attendance will be over 11,000. (M) 79. The life of a certain brand of car stereo is normally distributed with a mean of 5.2 years and a standard deviation of 0.9 year. What is the probability that a car stereo of this type chosen at random will last seven years or more? A. 0.9772 B. 0.0228 C. 0.228 D. 0.7190 E. 0.281 (S) 80. Martha manages a gift shop at a national park. Her daily revenues during the summer months follow a normal distribution with mean $2,400 and standard deviation $500. For a day chosen at random: (a) Find the probability that her revenue will be between $1,500 and $3,000. (b) Find the probability that her revenue will be under $3,600. (M) 81. The ages of applicants to an engineering school are normally distributed with a mean of 20.4 years and a standard deviation of 1.2 years. What is the probability that an applicant is younger than 18 years old or older than 22 years old? A. 0.0228 B. 0.098 C. 0.1146 D. 0.4427 E. 0.5573 (S) 82. The nightly attendance at the university summer outdoor theater series follows a normal distribution with mean 475 people and standard deviation 150 people. (a) Find the probability that on a night chosen at random the attendance will be greater than 175 people. (b) Find the probability that on a night chosen at random the attendance will be between 400 and 900 people. (M) 83. For a certain psychiatric evaluation, z scores above 2.18 and below 2.33 are considered abnormal. What is the probability that a person evaluated by this will be considered normal? A. 0.0245 B. 0.4878 C. 0.5122 D. 0.9755 E. 0.9714 (S) 84. Bill runs a bicycle repair shop. During the summer months the number of jobs per week follows a normal distribution with mean 45 and standard deviation 9. For a week chosen at random: (a) Find the probability that he will have 24 or more jobs. (b) Find the probability that he will have between 30 and 60 jobs.

Chapter 6 Normal Distributions 177 (S) 85. The number of visitors per day at a national park during the summer months follows a normal distribution with mean 10,500 and standard deviation 2750. For a summer day chosen at random: (a) Find the probability that the number of visitors will be between 6000 and 15,000. (b) Find the probability that the number of visitors will be less than 5000.