Test Bank: Introduction to Probability and Statistics

True/False Questions 1. The standard deviation of any normal random variable is always equal to one. Answer: False Type: Concept Difficulty: Easy 2. For any normal random variable, the probability that the random variable will equal one is always zero. Answer: True Type: Concept Difficulty: Medium 3. The graph of a standard normal random variable is always symmetric. Answer: True Type: Concept Difficulty: Easy 4. The formula will convert any normal distribution into the “standard normal distribution. ” Answer: True Type: Concept Difficulty: Easy 5.

Any normal random variable with standard deviation equal to one is a standard normal random variable. Answer: False Type: Concept Difficulty: Medium 6. The notation X – N(4, 32) indicates a normal distribution with mean 2 and standard deviation 3. Answer: False Type: Concept Difficulty: Easy 7. The total area under a normal curve is always equal to one. Answer: True Type: Concept Difficulty: Easy 8. The notation Z – N(0, 1) indicates a standard normal distribution. Answer: True Type: Concept Difficulty: Easy 9. The probability that a normal random variable will be within two standard deviations of its mean is approximately 0. 8. Answer: False Type: Concept Difficulty: Easy 10. The normal distribution is a continuous distribution. Answer: True Type: Concept Difficulty: Easy 11. The normal distribution can be used to approximate the binomial distribution when both np and n(1 – p) are at least five. Answer: True Type: Concept Difficulty: Easy 12. The normal distribution approximation to the binomial works best when n is large. Answer: True Type: Concept Difficulty: Easy 13. The formula can be used with both a normal and binominal distribution. Answer: True Type: Concept Difficulty: Easy Multiple Choice Questions 4. Find P(-2 < Z < 2). A)0. 9544 B)0. 4772 C)0. 9772 D)0. 6826 E)none of the above Answer: A Type: Computation Difficulty: Easy 15. Find P(-0. 5 < Z < 0. 5). A)0. 3830 B)0. 1915 C)0. 6515 D)0. 3085 E)none of the above Answer: A Type: Computation Difficulty: Easy 16. What is the probability that a standard normal variable will be between -0. 5 and 1. 00? A)0. 2857 B)0. 5328 C)0. 6687 D)0. 2500 E)none of the above Answer: B Type: Computation Difficulty: Easy 17. Find the probability that a standard normal random variable has a value greater than -1. 56. A)0. 0332 B)0. 0594 C)0. 9406

D)0. 9668 E)none of the above Answer: C Type: Computation Difficulty: Easy 18. Let X be a normally distributed random variable with mean 100 and standard deviation 20. Find two values, a and b, symmetric about the mean, such that the probability of the random variable being between them is 0. 99. A)90. 5, 105. 9 B)80. 2, 119. 8 C)22, 78 D)48. 5, 151. 5 E)90. 1, 109. 9 Answer: D Type: Computation Difficulty: Medium 19. A professor grades his students on a normal distribution, with mean at 75 and standard deviation of 15. If there are 39 students in his class, about how many score between 80 and 90? A)5

B)21 C)8 D)13 E)none of the above Answer: C Type: Computation Difficulty: Hard 20. A calculator manufacturer performs a test on its calculators and finds their working life to be normally distributed, with a mean of 2,150 hours and a standard deviation of 450 hours. What should the manufacturer advertise as the life of the calculators so that 90% of the calculators are covered? A)2,555 B)1,947 C)1,410 D)1,745 E)1,574 Answer: E Type: Computation Difficulty: Hard 21. You have two stocks: A and B. The price of each stock is normally distributed. Stock A has a mean of 25 and a standard deviation of 3.

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Stock B also has a mean of 25, but the standard deviation is 5. If I buy stock A at $25 and sell it on a randomly chosen day in the future (without knowing its price then), what is the probability that I will make at least $2 on each share? Answer the same for stock B. A)0. 2514, 0. 1554 B)0. 2514, 0. 3446 C)0. 2486, 0. 1554 D)0. 2486, 0. 3446 E)none of the above Answer: B Type: Computation Difficulty: Hard 22. Find two values symmetric around a mean of 20 such that they include an area equal to 0. 75. (standard deviation = 5). A)16. 65, 23. 35 B)19. 25, 20. 75 C)16. 25, 23. 75 D)14. 25, 25. 75

E)none of the above Answer: D Type: Computation Difficulty: Hard 23. A spark plug manufacturer believes that his plug lasts an average of 30,000 miles, with a standard deviation of 2,500 miles. What is the probability that a given spark plug of this type will last 37,500 miles before replacement? A)0. 0228 B)0. 0114 C)0. 0013 D)0. 0714 E)0. 0833 Answer: C Type: Computation Difficulty: Medium 24. Fluctuations in the exchange rate of dollars against the pound sterling over a short time period were approximated by a normal distribution with a mean of 2. 01 and a standard deviation of 0. 13.

What is the probability that the rate on a particular day was more than 1. 90? A)0. 8461 B)0. 3023 C)0. 8023 D)0. 3461 E)none of the above Answer: C Type: Computation Difficulty: Medium 25. The average time it takes for a letter in the United States to reach from one place in the 48 contiguous states to another is 3. 2 days, with a standard deviation of 0. 85 days. What is the probability of a letter arriving at its destination no more than four days after mailing? Assume a normal distribution. A)0. 1736 B)0. 3264 C)0. 8264 D)0. 6736 E)0. 6528 Answer: C Type: Computation Difficulty: Medium 6. The contents of a particular bottle of shampoo marked as 150 ml are found to be 153 ml at an average, with a standard deviation of 2. 5 ml. What proportion of shampoo bottles contain less than the marked quantity? Assume a normal distribution. A)0. 2192 B)0. 1151 C)0. 4452 D)0. 0548 E)none of the above Answer: B Type: Computation Difficulty: Medium 27. The age of people in a town is normally distributed, with a mean of 34 years and a standard deviation of 11 years. Find two values for age that will give a symmetric 0. 95 probability interval. A)28. 78, 39. 23 B)32. 30, 66. 30 C)23. 55, 44. 45

D)12. 44, 55. 56 E)15. 91, 51. 10 Answer: D Type: Computation Difficulty: Medium 28. The weight of apples in a farm is normally distributed, with a mean of 110 grams, and a standard deviation of 15 grams. Find the probability that an apple selected at random will weigh between 95 and 105 grams. A)0. 3413 B)0. 4706 C)0. 1293 D)0. 2108 E)0. 5294 Answer: D Type: Computation Difficulty: Medium 29. A grocery store has a mean accounts receivable of $264, with a standard deviation of $55. The accounts receivable are normally distributed. What proportion of all accounts will be greater than $275? A)0. B)0. 1 C)0. 4207 D)0. 0793 E)0. 0228 Answer: C Type: Computation Difficulty: Medium 30. A grocery store has a mean accounts receivable of $264, with a standard deviation of $55. The accounts receivable are approximately normally distributed. Find the value such that 45% of all the accounts exceed this value. That is, find x such that: P(X > x) = 0. 45. A)$257. 13 B)$354. 48 C)$270. 91 D)$309. 00 E)none of the above Answer: C Type: Computation Difficulty: Medium 31. The waist measurement of students in a college is normally distributed. The standard deviation is known to be five inches.

It is found that 15% of the students have waist sizes less than 28 inches. What proportion of students will have waists between 30 and 35 inches? A)0. 3795 B)0. 2389 C)0. 1406 D)0. 0983 E)none of the above Answer: A Type: Computation Difficulty: Hard 32. The IQs of the employees of a company are normally distributed, with a mean of 127 and a standard deviation of 11. What is the probability that the IQ of an employee selected at random will be between 120 and 130? A)0. 2389 B)0. 3453 C)0. 1064 D)0. 1325 E)0. 4638 Answer: B Type: Computation Difficulty: Medium 33.

The mean life of a computer disk drive is 2,000 hours, with a standard deviation of 140 hours. Assuming the life-time of the drives to be normally distributed, find the probability of a disk-drive lasting more than 1,800 hours? A)0. 4236 B)0. 9236 C)0. 8472 D)0. 5764 E)0. 2118 Answer: B Type: Computation Difficulty: Medium 34. The average bill for car repairs at a car service center is $196, with a standard deviation of $44. Assuming the bills to be normally distributed, find the probability of a bill exceeding $300. A)0. 4909 B)0. 0182 C)0. 9819 D)0. 1406 E)0. 0090 Answer: E Type: Computation Difficulty: Medium 35.

The GMAT scores of students in a college are normally distributed with a mean of 520 and a standard deviation of 41. What proportion of students have a score higher than 600? A)0. 9744 B)0. 2372 C)0. 4774 D)0. 0255 E)none of the above Answer: D Type: Computation Difficulty: Medium 36. The probability that a normal random variable with mean zero and standard deviation one will equal the number 1. 00 is: A)1 B)0. 9 C)0. 3413 D)0. 1587 E)0 Answer: E Type: Concept Difficulty: Medium 37. Suppose that X is a normal random variable with mean 17 and standard deviation 10. The probability that the value of X will be between -2. and 36. 6 is: A)0 B)0. 90 C)a number very close to 1 D)0. 95 E)0. 99 Answer: D Type: Computation Difficulty: Medium 38. Suppose that X is a normal random variable with mean 10 and standard deviation 4. Then the probability that X will be greater than 12 is: A)0. 1587 B)0. 3085 C)0. 1915 D)0. 4772 E)none of the above Answer: B Type: Computation Difficulty: Medium 39. A normal random variable has a distribution that is: A)always symmetric B)never symmetric C)sometimes symmetric D)symmetric if the mean is positive E)symmetric if the variance is negative Answer: A Type: Concept Difficulty: Medium 0. The distribution of X, the number of cars sold per day, where X can be 0, 1, 2, 3, 4, or 5 is: A)sometimes normally distributed B)never normal C)always normal D)a uniform distribution E)none of the above Answer: B Type: Concept Difficulty: Medium 41. What is the probability that a normal random variable with mean 15 and standard deviation 5 will have a value of exactly 25? A)0. 0228 B)0. 0456 C)0. 9772 D)0 E)1 Answer: D Type: Concept Difficulty: Medium 42. If X is a normal random variable with mean 12 and standard deviation 2, then the probability that X will exceed 16 is? A)0. 4772 B)0. 0228 C)0. 9772 D)0 E)1 Answer: B Type: Computation Difficulty: Medium 43. If X is a normal random variable with mean 15 and standard deviation 10, then the probability that X will have a negative value is: A)0. 0668 B)0. 432 C)0. 9332 D)0. 8664 E)none of the above Answer: A Type: Computation Difficulty: Medium 44. If X is a normally distributed random variable with mean 16 and variance 64, the probability that the random variable will have a value between 0. 32 and 31. 68 is: A)0. 99 B)0. 90 C)0. 85 D)1 E)0. 95 Answer: E Type: Computation Difficulty: Medium 45.

For a normally distributed random variable with mean zero and standard deviation five, the probability that its value will be greater than -5 is: A)0. 4772 B)0. 9544 C)0. 3413 D)0. 8413 E)none of the above Answer: D Type: Computation Difficulty: Medium 46. What is the probability that a standard normal random variable is between -0. 4 and 1. 4? A)0. 3413 B)0. 4254 C)0. 5746 D)0. 2638 E)none of the above Answer: C Type: Computation Difficulty: Easy 47. All of the following are characteristics of the normal distribution, except: A)symmetric about the mean B)bell-shaped curve C)total area under the curve is always one

D)it is a discrete distribution E)probability that x is equal to any specific value is zero Answer: D Type: Computation Difficulty: Medium 48. Find two values symmetric about a mean of 100, standard deviation of 10, such that they include an area equal to 0. 95. A)90, 110 B)80. 4, 119. 6 C)98. 04, 101. 96 D)70, 130 E)none of the above Answer: B Type: Computation Difficulty: Medium 49. A tire manufacturer believes its tires will last an average of 48,000 miles, with standard deviation of 2,000 miles. What is the probability that one of these tires, chosen at random, will last at least 50,000 miles?

A)0. 6587 B)0. 3413 C)0. 1587 D)0. 4772 E)none of the above Answer: C Type: Computation Difficulty: Medium 50. Suppose that an instructor gives an exam. This instructor wants to give those students in the top 2. 5% an A on this exam. What will the cutoff be for an A, if the average score on this exam is 80, with a standard deviation of 5? A)about 80 B)about 90 C)about 85 D)about 86 E)none of the above Answer: B Type: Computation Difficulty: Hard Use the following to answer questions 51-54: LittleAir operates a fleet of regional jets on a contract basis for a major air carrier.

LittleAir’s jets seat only 50 passengers, but because passengers’ travel plans often change, LittleAir books up to 60 reservations for a typical flight. Booked passengers have a “no-show” probability of 0. 25. 51. Suppose LittleAir loses money if the number of passengers on a flight is less than 40. What is the probability that a randomly selected LittleAir flight will have fewer than 40 passengers? A)0. 0367 B)0. 0505 C)0. 0681 D)0. 0901 E)0. 1492 Answer: B Type: Computation Difficulty: Medium 52. What is the probability that a randomly selected LittleAir flight will be overbooked (i. . , have more passengers show up than there are seats available)? A)0. 1170 B)0. 0901 C)0. 0681 D)0. 0505 E)0. 0367 Answer: D Type: Computation Difficulty: Medium 53. What is the probability that a randomly selected LittleAir flight will be full? A)0. 1170 B)0. 0901 C)0. 0681 D)0. 0505 E)0. 0367 Answer: B Type: Computation Difficulty: Medium 54. Suppose LittleAir gives compensation vouchers to any passenger who is denied a seat on an overbooked flight. Because these vouchers are valuable (> $200), management would like to keep the number of them on-hand at a minimum.

How many vouchers should be held at the gate such that there are enough for at least 99% of all situations? A)1 voucher B)2 vouchers C)3 vouchers D)4 vouchers E)5 vouchers Answer: C Type: Computation Difficulty: Hard Use the following to answer questions 55-57: The ski season at a popular resort destination lasts 120 days. Experience has shown that the probability of snow on any given day is 0. 55 and is independent of whether or not there was snow on the previous day. 55. What is the probability of there being more than 60 days of snow in any given year? A)0. 7967 B)0. 8438 C)0. 8643

D)0. 8830 E)0. 8997 Answer: B Type: Computation Difficulty: Medium 56. What is the probability of there being fewer than 55 days of snow in any given year? A)0. 0409 B)0. 0268 C)0. 0217 D)0. 0174 E)0. 0139 Answer: D Type: Computation Difficulty: Medium 57. Suppose a particular hotel at this destination breaks even or makes money so long as there are at least 50 days of snow but no more than 70 days of snow. What is the probability of the hotel’s losing money in any given year? A)0. 8438 B)0. 7955 C)0. 7944 D)0. 7664 E)0. 7657 Answer: B Type: Computation Difficulty: Hard 58.

If, for a binomially distributed random variable n*p = 5 and n*(1-p) = 5, then a _____________ distribution with a mean equal to _____ and a standard deviation equal to _____ typically can be used. A)Normal; ; B)Normal; ; C)Exponential; ; D)Exponential; ; E)Hypergeometric; ; Answer: A Type: Concept Difficulty: Easy Short Answer Questions Use the following to answer questions 59-67: If x ~ N(40, 32): 59. Find p(X ; 37) Answer: 0. 8413 Type: Computation Difficulty: Medium 60. Find p(X ; 47) Answer: 0. 0099 Type: Computation Difficulty: Medium 61. Find p(42 ; X ; 47) Answer: 0. 415 Type: Computation Difficulty: Medium 62. Find p(X ; 41) Answer: 0. 3707 Type: Computation Difficulty: Medium 63. Find p(36 ; X ; 41) Answer: 0. 5375 Type: Computation Difficulty: Medium 64. Find p(36 ; X ; 39) Answer: 0. 2789 Type: Computation Difficulty: Medium 65. Find x1 such that: p(X ; xl) = 0. 0475 Answer: 45 Type: Computation Difficulty: Medium 66. Find xl such that: p(40 ; X ; xl) = 0. 3770 Answer: 43. 48 Type: Computation Difficulty: Medium 67. Find x1 such that: p(X ; x1) = 0. 0154 Answer: 33. 52 Type: Computation Difficulty: Medium

Use the following to answer questions 68-69: There are two shipping routes between a plant and a distribution point. The average time by route A is 220 minutes with a standard deviation of 20 minutes. The average time and standard deviation by route B are 200 and 40, respectively. Assume the distributions of trips can be approximated by normal curves. 68. What proportion of route B trips takes longer than the average route A trip? Answer: 0. 3085 Type: Computation Difficulty: Medium 69. 95% of route A trips take between what two values that are equidistant from the mean? Answer: [180. 8, 259. 2] Type: Computation Difficulty: Medium

Use the following to answer questions 70-73: The amount dispensed into bottles by a machine in a ketchup plant is supposed to be normally distributed with a mean of 10 ounces and a standard deviation of 0. 5 ounce. If the machine is working properly, what is the probability that a single bottle chosen at random from the assembly line will have: 70. More than 11 ounces or less than 9. 5 ounces? Answer: 0. 1815 Type: Computation Difficulty: Medium 71. Between 9. 5 and 11 ounces? Answer: 0. 8185 Type: Computation Difficulty: Medium 72. What would you think if a single bottle chosen at random had less than . 5 ounces? Answer: p(x ; 8. 5) = 0. 0013, so we might conclude the machine is operating improperly. Type: Computation Difficulty: Medium 73. Between what two values symmetric about the mean would you expect to find 99% of the bottles filled by the machine, if it is operating properly? Answer: [8. 712, 11. 288] Type: Computation Difficulty: Medium 74. If the contents of bottles coming off a production line are normally distributed with a mean of 16 ounces and a variance of 0. 625, what’s the probability of choosing a bottle at random and finding its contents to be less than 15. 1 ounces? Answer: 0. 2676 Type: Computation Difficulty: Medium 75. A seed packet says that 90% of lettuce seedlings should be between 2 and 2. 5 inches high after 5 weeks. Assuming an average height of 2. 25 inches (and a normal distribution), what’s the standard deviation of heights of 5-week-old seedlings? Answer: 0. 1520 Type: Computation Difficulty: Medium Use the following to answer questions 76-78: If the distribution of heights of mature poppy plants is normal, with a mean of 16 inches and a standard deviation of 3 inches, what proportion of the poppies will be: 76.

Between 10 and 20 inches? Answer: 0. 8854 Type: Computation Difficulty: Medium 77. Less than 9 inches? Answer: 0. 0099 Type: Computation Difficulty: Medium 78. More than 24 inches? Answer: 0. 0038 Type: Computation Difficulty: Medium 79. You are interested in the incomes of your customers. A random sample of customer incomes yields a mean income of $35,000 with a standard deviation of $4,421. A) Determine what percent of the population would have a salary above $38,000. B) What income range that is symmetric about the mean would include 95% of your customers? Answer:

A) 24. 83% have incomes above $35,000. B) 95% of your customers have an income between $26,330 and $43,670. Type: Computation Difficulty: Medium 80. Half of all mutual funds of a particular class charge up-front administration fees. Assuming that a random sample of 60 of these mutual funds is taken, calculate: A) The mean and standard deviation of the normal approximation of the binomial. B) The probability that no more than 40 of the mutual funds sampled charge an up-front administration fee. Answer: A) Mean = 30, standard deviation = 3. 873 B) Prob(# charging fee = 40) = 0. 9966

Type: Computation Difficulty: Medium 81. The owner of a 100-room hotel has discovered that his reservations team has booked 110 reservations for an upcoming weekend. Experience has shown that 10% of reservations are “no shows. ” How likely is this hotel to be overbooked (i. e. , have more guests arrive than there are rooms available) for this particular weekend? Answer: 0. 3156 Type: Computation Difficulty: Medium 82. Harry Highroller likes to bet on the roulette wheel when he is in Las Vegas. Roulette wheels in Las Vegas typically have 38 spaces: 18 of them are red; 18 are black; and 2 are green.

Harry’s “strategy” is simple: He bets $2 on every spin, he always bets on red, and he always plays exactly 100 spins. If red comes up, Harry wins $2. If either black or green comes up, Harry loses $2. Suppose Harry has decided to play his usual strategy tonight. A) What are the mean and standard deviation of the normal approximation of the binomial in this instance? B) What is the probability that, after 100 plays, Harry will be ahead (i. e. , have more money than he started with)? Answer: A) Mean = 47. 37 and standard deviation = 4. 993 B) 0. 2643 Type: Computation Difficulty: Medium

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