Analytical Methods for Business (University of Arizona)

ch06

*Student:*

1. A continuous random variable is characterized by uncountable values and can take on any value within an interval.

True False

2. We are often interested in finding the probability that a continuous random variable assumes a particular value.

True False

3. The probability density function of a continuous random variable can be regarded as a counterpart of the probability mass function of a discrete random variable.

True False

4. Cumulative distribution functions can only be used to compute probabilities for continuous random variables.

True False

5. The continuous uniform distribution describes a random variable, defined on the interval [*a*, *b*], that has an equally likely chance of assuming values within any subinterval of [*a*, *b*] with the same length.

True False

6. The probability density function of a continuous uniform distribution is positive for all values between –**∞** and +**∞**.

True False

7. The mean of a continuous uniform distribution is simply the average of the upper and lower limits of the interval on which the distribution is defined.

True False

8. The mean and standard deviation of the continuous uniform distribution are equal. True False

9. The probability density function of a normal distribution is in general characterized by being symmetric and bell-shaped.

True False

10. Examples of random variables that closely follow a normal distribution include the age and the class year designation of a college student.

True False

11. Given that the probability distribution is normal, it is completely described by its mean **μ** > 0 and its standard deviation **σ** > 0.

True False

12. Just as in the case of the continuous uniform distribution, the probability density function of the normal distribution may be easily used to compute probabilities.

True False

13. The standard normal distribution is a normal distribution with a mean equal to zero and a standard deviation equal to one.

True False

lOMoARcPSD|3013804

14. The letter *Z* is used to denote a random variable with any normal distribution. True False

15. The standard normal table is also referred to as the *z* table. True False

16. Which of the following is correct?

A. A continuous random variable has a probability density function but not a cumulative distribution function.

B. A discrete random variable has a probability mass function but not a cumulative distribution function. C.A continuous random variable has a probability mass function, and a discrete random variable has a

probability density function.

D.A continuous random variable has a probability density function, and a discrete random variable has a probability mass function.

17. Which of the following does not represent a continuous random variable?

A. Height of oak trees in a park.

B. Heights and weights of newborn babies.

C. Time of a flight between Chicago and New York.

D. The number of customer arrivals to a bank between 10 am and 11 am.

18. Which of the following is *not* a characteristic of a probability density function *f*(*x*)?

A. *f*(*x*) **≥** 0 for all values of* x*.

B. *f*(*x*) is symmetric around the mean.

C. The area under *f*(*x*) over all values of *x* equals one.

D. *f*(*x*) becomes zero or approaches zero if* x *increases to +infinity or decreases to -infinity.

19. The cumulative distribution function is denoted and defined as which of the following?

A. *f*(*x*) and* f*(*x*) =* P*(*X ***≤*** x*)

B. *f*(*x*) and* f*(*x*) =* P*(*X ***≥*** x *)

C. *F*(*x*) and* F*(*x*) =* P*(*X ***≤*** x*)

D. *F*(*x*) and* F*(*x*) =* P*(*X ***≥*** x*)

20. The cumulative distribution function *F*(*x*) of a continuous random variable *X* with the probability density function *f*(*x*) is which of the following?

A. The area under *f* over all values *x*

B. The area under *f* over all values that are *x* or less

C. The area under *f* over all values that are *x* or more

D. The area under *f* over all non-negative values that are x or less

21. A continuous random variable has the uniform distribution on the interval [*a*, *b*] if its probability density

function *f*(*x*) .

A. Is symmetric around its mean

B. Is bell-shaped between *a* and *b*

C. Is constant for all *x* between *a* and *b*, and 0 otherwise

D. Asymptotically approaches the *x* axis when *x* increases to +**∞** or decreases to –**∞**

22. The height of the probability density function *f*(*x*) of the uniform distribution defined on the interval [*a, b*]

is .

A. 1/(*b* – *a*) between *a* and *b*, and zero otherwise

B. (*b* – *a*)/2 between *a* and *b*, and zero otherwise

C. (*a* + *b*)/2 between *a* and *b*, and zero otherwise

D. 1/(*a* + *b*) between *a* and *b*, and zero otherwise

lOMoARcPSD|3013804

23. The waiting time at an elevator is uniformly distributed between 30 and 200 seconds. Find the mean and standard deviation of the waiting time.

A. 115 seconds and 49.07 seconds

B. 1.15 minutes and 0.4907 minutes

C. 1.15 minutes and 24.08333 (minute)2

D. 115 seconds and 2408.3333 (second)2

24. The waiting time at an elevator is uniformly distributed between 30 and 200 seconds. What is the probability a rider must wait between 1 minute and 1.5 minutes?

A. 0.1765

B. 0.3529

C. 0.5294

D. 0.8824

25. The waiting time at an elevator is uniformly distributed between 30 and 200 seconds. What is the probability a rider must wait more than 1.5 minutes?

A. 0.3529

B. 0.4500

C. 0.5294

D. 0.6471

26. The waiting time at an elevator is uniformly distributed between 30 and 200 seconds. What is the probability a rider waits less than two minutes?

A. 0.4706

B. 0.5294

C. 0.6000

D. 0.7059

27. The time of a call to a technical support line is uniformly distributed between 2 and 10 minutes. What are the mean and variance of this distribution?

A. 6 minutes and 2.3094 (minutes)2

B. 6 minutes and 5.3333 (minutes)2

C. 6 minutes and 5.3333 minutes

D. 8 minutes and 2.3094 minutes

28. An analyst is forecasting net income for Excellence Corporation for the next fiscal year. Her low-end estimate of net income is $250,000, and her high-end estimate is $350,000. Prior research allows her to

assume that net income follows a continuous uniform distribution. The probability that net income will be

greater than or equal to $337,500 is .

A. 12.5%

B. 29.6%

C. 87.5%

D. 96.4%

29. Alex is in a hurry to get to work and is rushing to catch the bus. She knows that the bus arrives every six minutes during rush hour, but does not know the exact times the bus is due. She realizes that from the time she arrives at the stop, the amount of time that she will have to wait follows a uniform distribution with a lower bound of 0 minutes and an upper bound of six minutes. What is the probability that she will have to wait more than two minutes?

A. 0.1667

B. 0.3333

C. 0.6667

D. 1.0000

lOMoARcPSD|3013804

30. Suppose the average price of gasoline for a city in the United States follows a continuous uniform distribution with a lower bound of $3.50 per gallon and an upper bound of $3.80 per gallon. What is the probability a randomly chosen gas station charges more than $3.70 per gallon?

A. 0.3000

B. 0.3333

C. 0.6667

D. 1.0000

31. How many parameters are needed to fully describe any normal distribution?

A. 1

B. 2

C. 3

D. 4

32. What does it mean when we say that the tails of the normal curve are asymptotic to the *x* axis?

A. The tails get closer and closer to the *x* axis but never touch it.

B. The tails gets closer and closer to the *x* axis and eventually touch it.

C. The tails get closer and closer to the *x* axis and eventually cross this axis.

D. The tails get closer and closer to the *x* axis and eventually become this axis.

33. The probability that a normal random variable is less than its mean is .

A. 0.0

B. 0.5

C. 1.0

D. Cannot be determined

34. Let *X* be normally distributed with mean **μ** and standard deviation **σ** > 0. Which of the following is true about the *z* value corresponding to a given *x* value?

A. A positive *z* = (*x* – **μ**)/**σ** indicates how many standard deviations *x* is above **μ***.*

B. A negative *z* = (*x* – **μ**)/**σ** indicates how many standard deviations *x* is below **μ***.*

C. The *z* value corresponding to *x* = **μ** is zero.

D. All of the above.

35. It is known that the length of a certain product *X* is normally distributed with **μ** = 20 inches. How is the

probability related to ?

A. is greater than .

B. is smaller than .

C. is the same as .

D. No comparison can be made with the given information.

36. It is known that the length of a certain product *X* is normally distributed with **μ** = 20 inches. How is the

probability related to ?

A. is greater than .

B. is smaller than .

C. is the same as .

D. No comparison can be made with the given information.

37. It is known that the length of a certain product *X* is normally distributed with **μ** = 20 inches. How is the

probability related to ?

A. is greater than .

B. is smaller than .

C. is the same as .

D. No comparison can be made with the given information.

lOMoARcPSD|3013804

38. It is known that the length of a certain product *X* is normally distributed with **μ** = 20 inches and **σ** = 4

inches. How is the probability related to ?

A. is greater than .

B. is smaller than .

C. is the same as .

D. No comparison can be made with the given information.

39. The probability *P*(*Z* < -1.28) is closest to .

A. -0.10

B. 0.10

C. 0.20

D. 0.90

40. The probability *P*(*Z* > 1.28) is closest to .

A. -0.10

B. 0.10

C. 0.20

D. 0.90

41. Find the probability *P*(-1.96 **≤** *Z* **≤** 0).

A. 0.0250

B. 0.0500

C. 0.4750

D. 0.5250

42. Find the probability *P*(-1.96 **≤** *Z* **≤** 1.96).

A. 0.0500

B. 0.9500

C. 0.9750

D. 1.9500

43. Find the *z* value such that .

A. *z *= -1.33

B. *z *= 0.1814

C. *z *= 0.8186

D. *z *= 1.33

44. Find the *z* value such that .

A. *z *= -1.645

B. *z *= -1.96

C. *z *= 1.645

D. *z *= 1.96

45. You work in marketing for a company that produces work boots. Quality control has sent you a memo detailing the length of time before the boots wear out under heavy use. They find that the boots wear out in an average of 208 days, but the exact amount of time varies, following a normal distribution with a standard deviation of 14 days. For an upcoming ad campaign, you need to know the percent of the pairs that last longer than six months**—**that is, 180 days. Use the empirical rule to approximate this percent.

A. 2.5%

B. 5%

C. 95%

D. 97.5%

lOMoARcPSD|3013804

46. A hedge fund returns on average 26% per year with a standard deviation of 12%. Using the empirical rule, approximate the probability the fund returns over 50% next year.

A. 0.5%

B. 1%

C. 2.5%

D. 5%

47. For any normally distributed random variable with mean **μ** and standard deviation **σ**, the percent of the

observations that fall between and is *closest* to .

A. 68%

B. 68.26%

C. 95%

D. 95.44%

48. For any normally distributed random variable with mean **μ** and standard deviation **σ**, the proportion of the

observations that fall outside the interval [**μ** – **σ**, **μ** + **σ**] is *closest* to .

A. 0.0466

B. 0.3174

C. 0.8413

D. 0.1687

49. Sarah’s portfolio has an expected annual return at 8%, with an annual standard deviation at 12%.

If her investment returns are normally distributed, then in any given year Sarah has approximately

.

A. A 50% chance that the actual return will be greater than 8%

B. About a 68% chance that the actual return will fall within 4% and 20%

C. About a 68% chance that the actual return will fall within -20% and 20%

D. About a 95% chance that the actual return will fall within -4% and 28%.

50. If *X* has a normal distribution with and , then the probability can be

expressed in terms of a standard normal variable *Z* as .

A.

B.

C.

D.

51. The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take more than five hours to construct a soapbox derby car.

A. 0

B. 0.0228

C. 0.4772

D. 0.9772

52. The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car.

A. 0.3085

B. 0.3830

C. 0.6170

D. 0.6915

lOMoARcPSD|3013804

53. The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take exactly 3.7 hours to construct a soapbox derby car.

A. 0.0000

B. 0.5000

C. 0.7580

D. 0.2420

54. Let *X* be normally distributed with mean *µ* = 250 and standard deviation **σ** = 80. Find the value *x* -such that *P*(*X* **≤** *x*) = 0.0606.

A. -1.55

B. 1.55

C. 126

D. 374

55. Let *X* be normally distributed with mean *µ* = 250 and standard deviation **σ** = 80. Find the value *x* such that *P*(*X* **≤** *x*) = 0.9394.

A. -1.55

B. 1.55

C. 126

D. 374

56. Let *X* be normally distributed with mean *µ* = 25 and standard deviation **σ** = 5. Find the value *x* such that *P*(*X ***≥*** x*) = 0.1736.

A. -0.94

B. 0.94

C. 20.30

D. 29.70

57. The salary of teachers in a particular school district is normally distributed with a mean of $50,000 and a standard deviation of $2,500. Due to budget limitations, it has been decided that the teachers who are in the top 2.5% of the salaries would not get a raise. What is the salary level that divides the teachers into one group that gets a raise and one that doesn’t?

A. -1.96

B. 1.96

C. 45,100

D. 54,900

58. The starting salary of an administrative assistant is normally distributed with a mean of $50,000 and a standard deviation of $2,500. We know that the probability of a randomly selected administrative assistant making a salary between **μ** – *x* and **μ** + *x* is 0.7416. Find the salary range referred to in this statement.

A. $42,825 to $52,825

B. $42,825 to $57,175

C. $47,175 to $52,825

D. $47,175 to $57,175

59. An investment consultant tells her client that the probability of making a positive return with her suggested portfolio is 0.90. What is the risk, measured by standard deviation, that this investment manager has assumed in his calculation if it is known that returns from her suggested portfolio are normally distributed with a mean of 6%?

A. 1.28%

B. 4.69%

C. 6.00%

D. 10.0%

lOMoARcPSD|3013804

60. The stock price of a particular asset has a mean and standard deviation of $58.50 and $8.25, respectively. Use the normal distribution to compute the 95th percentile of this stock price.

A. -1.645

B. 1.645

C. 44.93

D. 72.07

61. **EXHIBIT** **6-1. **You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough. The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F.

Refer to Exhibit 6-1. One sleeping bag you are considering advertises that it is good for temperatures down to 25°F. What is the probability that this bag will be warm enough on a randomly selected May night at the park?

A. 0.1894

B. 0.3106

C. 0.8106

D. 0.8800

62. **EXHIBIT** **6-1. **You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough. The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F.

Refer to Exhibit 6-1. An inexpensive bag you are considering advertises to be good for temperatures down to 38°F. What is the probability that the bag will *not* be warm enough?

A. 0.2266

B. 0.2734

C. 0.7500

D. 0.7734

63. **EXHIBIT** **6-1. **You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough. The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F.

Refer to Exhibit 6-1. Above what temperature must the sleeping bag be suited such that the temperature will be too cold only 5% of the time?

A. -1.645

B. 1.645

C. 18.84

D. 45.16

64. **EXHIBIT** **6-2. **Gold miners in Alaska have found, on average, 12 ounces of gold per 1000 tons of dirt excavated with a standard deviation of 3 ounces. Assume the amount of gold found per 1000 tons of dirt is normally distributed.

Refer to Exhibit 6-2. What is the probability the miners find more than 16 ounces of gold in the next 1000 tons of dirt excavated?

A. 0.0918

B. 0.4082

C. 0.5918

D. 0.9082

lOMoARcPSD|3013804

65. **EXHIBIT** **6-2. **Gold miners in Alaska have found, on average, 12 ounces of gold per 1000 tons of dirt excavated with a standard deviation of 3 ounces. Assume the amount of gold found per 1000 tons of dirt is normally distributed.

Refer to Exhibit 6-2. What is the probability the miners find between 10 and 14 ounces of gold in the next 1000 tons of dirt excavated?

A. 0.2514

B. 0.4972

C. 0.5028

D. 0.7486

66. **EXHIBIT** **6-2. **Gold miners in Alaska have found, on average, 12 ounces of gold per 1000 tons of dirt excavated with a standard deviation of 3 ounces. Assume the amount of gold found per 1000 tons of dirt is normally distributed.

Refer to Exhibit 6-2. If the miners excavated 1000 tons of dirt, how little gold must they have found such that they find that amount or less only 15% of the time?

A. -1.04

B. 1.04

C. 8.88

D. 15.12

67. Suppose the life of a particular brand of laptop battery is normally distributed with a mean of 8 hours and a standard deviation of 0.6 hours. What is the probability that the battery will last more than 9 hours before running out of power?

A. 0.0475

B. 0.4525

C. 0.9525

D. 1.6667

68. A superstar major league baseball player just signed a new deal that pays him a record amount of money. The star has driven in an average of 110 runs over the course of his career, with a standard deviation

of 31 runs. An average player at his position drives in 80 runs. What is the probability the superstar bats in fewer runs than an average player next year? Assume the number of runs batted in is normally distributed.

A. 0.1660

B. 0.3340

C. 0. 8340

D. 0.9700

69. If an exponential distribution has the rate parameter **λ** = 5, what is its expected value?

A. 5

B. 1/5

C. 1/25

D. 5/2

70. If an exponential distribution has the rate parameter **λ** = 5, what is its variance?

A. 5

B. 1/5

C. 1/25

D. 5/2

71. What can be said about the expected value and standard deviation of an exponential distribution?

A. The expected value is equal to the standard deviation.

B. The expected value is equal to the square of the standard deviation.

C. The expected value is equal to the reciprocal of the standard deviation.

D. The expected value is equal to the square root of the standard deviation.

lOMoARcPSD|3013804

72. Let the time between two consecutive arrivals at a grocery store check-out line be exponentially distributed with a mean of three minutes. Find the probability that the next arrival does not occur until at least four minutes have passed since the last arrival.

A. 0.0000

B. 0.2636

C. 0.4724

D. 0.7364

73. **EXHIBIT** **6-3. **Patients scheduled to see their primary care physician at a particular hospital wait, on average, an additional eight minutes after their appointment is scheduled to start. Assume the time that patients wait is exponentially distributed.

Refer to Exhibit 6-3. What is the probability a randomly selected patient will have to wait more than 10 minutes?

A. 0.2865

B. 0.4493

C. 0.5507

D. 0.7135

74. **EXHIBIT** **6-3. **Patients scheduled to see their primary care physician at a particular hospital wait, on average, an additional eight minutes after their appointment is scheduled to start. Assume the time that patients wait is exponentially distributed.

Refer to Exhibit 6-3. What is the probability a randomly selected patient will see the doctor within five minutes of the scheduled time?

A. 0.2019

B. 0.4647

C. 0.5353

D. 0.7981

75. **EXHIBIT** **6-4. **The average time between trades for a high-frequency trading investment firm is 40 seconds. Assume the time between trades is exponentially distributed.

Refer to Exhibit 6-4. What is the probability that the time between trades for a randomly selected trade and the one proceeding it is less than 20 seconds?

A. 0.1354

B. 0.3935

C. 0.6065

D. 0.8446

76. **EXHIBIT** **6-4. **The average time between trades for a high-frequency trading investment firm is 40 seconds. Assume the time between trades is exponentially distributed.

Refer to Exhibit 6-4. What is the probability that the time between trades for a randomly selected trade and the one proceeding it is more than a minute?

A. 0.2231

B. 0.4869

C. 0.5134

D. 0.7769

77. If has a lognormal distribution, what can be said of the distribution of the random variable *X*?

A. *X *follows a normal distribution.

B. *X *follows an exponential distribution.

C. *X *follows a standard normal distribution.

D. *X *follows a continuous uniform distribution.

lOMoARcPSD|3013804

78. Find the mean of the lognormal variable if the mean and standard deviation of the underlying normal variable are 2 and 0.8, respectively.

A. 0.69

B. 2.32

C. 10.18

D. 11.02

79. Find the variance of the lognormal variable if the mean and variance of the underlying normal variable are 2 and 1, respectively.

A. 0

B. 12.18

C. 15.97

D. 255.02

80. **EXHIBIT** **6-5. **The mean travel time to work is 25.2 minutes (U.S. Census 2010). Further, suppose that commute time follows a log-normal distribution with a standard deviation of 10 minutes.

Refer to Exhibit 6-5. What is the probability a randomly selected U.S. worker has a commute time of more than half an hour?

A. 25.78%

B. 31.56%

C. 68.44%

D. 74.22%

81. **EXHIBIT** **6-5. **The mean travel time to work is 25.2 minutes (U.S. Census 2010). Further, suppose that commute time follows a log-normal distribution with a standard deviation of 10 minutes.

Refer to Exhibit 6-5. What is the probability a randomly selected U.S. worker has a commute time of less than 20 minutes?

A. 30.15%

B. 34.09%

C. 65.91%

D. 69.85%

82. **EXHIBIT** **6-6. **Let the lifetime of a new Jet Ski be represented by a lognormal variable, where *X* is normally distributed. Let the mean of the lifetime of the Jet Ski be six years with a standard deviation of three years.

Refer to Exhibit 6-6. What proportion of the Jet Skis will last less than seven years?

A. 0.2877

B. 0.3707

C. 0.6293

D. 0.7123

83. **EXHIBIT** **6-6. **Let the lifetime of a new Jet Ski be represented by a lognormal variable, where *X* is normally distributed. Let the mean of the lifetime of the Jet Ski be six years with a standard deviation of three years.

Refer to Exhibit 6-6. What proportion of the Jet Skis will last nine or more years?

A. 0.1379

B. 0.1587

C. 0.8413

D. 0.8621

lOMoARcPSD|3013804

84. When attending a movie, patrons are interested in avoiding the pre-movie trivia games, ads, and previews. It is known that the previews begin at the scheduled movie start time and they last between 5 and 15 minutes. Assume that the time of the previews is uniformly distributed.

a. Find the expected time and variance of the movie preview duration.

b. What is the probability that on a given day the previews last between 10 and 12 minutes?

85. Snack food companies always have a target weight when filling a box of snack crackers. It is not possible, however, to always fill boxes to the exact target weight. For a particular box of crackers, the target weight is 14 ounces. The filling machine drops between 13.5 and 15 ounces of crackers into each box. If these weights are uniformly distributed, what is the expected value of the fill weights? Does this value match the target weight? If not, what impact does this difference have on the manufacturer’s costs to produce these crackers?

86. Snack food companies always have a target weight when filling a box of snack crackers. It is not possible, however, to always fill boxes to the exact target weight. For a particular box of crackers, the target weight is 14 ounces. The filling machine drops between 13.5 and 15 ounces of crackers into each box. Let these weights be uniformly distributed.

a. What is the probability that the weight in a box will be less than 14 ounces?

b. What is the probability that the weight in a box will be between 14.5 and 15.5 ounces?

87. The time you must wait for an Orange Line train of the Massachusetts Bay Transit Authority follows a uniform distribution with a lower bound of 0 minutes and an upper bound of 8 minutes. Jonathan is running to catch the train in order to get to a meeting. He knows that the train needs to arrive within five minutes or else he will be late. What is the probability that he will be late to his meeting?

lOMoARcPSD|3013804

88. Suppose Jennifer is waiting for a taxi cab. A taxi cab’s arrival time is equally likely at any constant time range in the next 12 minutes.

a. Calculate the expected arrival time.

b. What is the probability that a taxi arrives in three minutes or less?

89. Find the following probabilities for a standard normal random variable *Z*.

a. b.

c. d.

90. Find the following probabilities for a standard normal random variable *Z*.

a. b.

c. d.

91. Find the value of *z* for which the standard normal random variable *Z* satisfies the following:

a. b.

c. d.

lOMoARcPSD|3013804

92. Given normally distributed random variable *X* with a mean of 10 and a variance of 4, find the following probabilities.

a.

b.

c.

d.

93. Given normally distributed random variable *X* with a mean of 12 and a standard deviation of 3.4, find the following probabilities.

a. b. c. d.

94. A normal random variable *X* has a mean of 17 and a variance of 5.

a. Find the value *x* for which *P*(*X* **≤** *x*) = 0.0020.

b. Find the value of *x* for which *P*(*X* > *x*) = 0.0122.

95. A soft drink company fills two-liter bottles on several different lines of production equipment. The fill volumes are normally distributed with a mean of 1.97 liters and a variance of 0.04 (liter)2.

a. Find the probability that a randomly selected two-liter bottle would contain between 1.95 and 2.03 liters.

b. If *X* is the fill volume of a randomly selected two-liter bottle, find the value of *x* for which *P*(*X* > *x*) = 0.3228.

lOMoARcPSD|3013804

96. A soft drink company fills two-liter bottles on several different lines of production equipment. The fill volumes are normally distributed with a mean of 1.97 liters and a variance of 0.04 (liter)2.

a. Find the probability that a randomly selected two-liter bottle would contain more than 1.92 liters.

b. If *X* is the fill volume of a randomly selected two-liter bottle, find the value of *x* for which *P*(*X* < *x*) = 0.6293.

97. You are considering the risk-return profile of two mutual funds for investment. The relatively risky fund promises an expected return of 9%, with a standard deviation of 12%. The relatively less risky fund promises an expected return and standard deviation of 5% and 8%, respectively.

a. Which mutual fund will you pick if your objective is to minimize the probability of earning a negative return?

b. Which mutual fund will you pick if your objective is to maximize the probability of earning a return between 8% and 12%?

98. The annual return of a well-known mutual fund has historically had a mean of about 10% and a standard deviation of 21%. Suppose the return for the following year follows a normal distribution, with the historical mean and standard deviation. What is the probability that you will lose money in the next year by investing in this fund?

99. The East Los Angeles Interchange is the busiest freeway interchange in the world. In 2008, an average of 550,000 cars passed through the intersection per day with a standard deviation of 100,000. What is the probability more than 620,000 use the interchange on a random day? Assume the number of cars on the interchange is approximately normally distributed.

lOMoARcPSD|3013804

100.The weight of competition pumpkins at the Circleville Pumpkin Show in Circleville, Ohio, can be represented by a normal distribution with a mean of 703 pounds and a standard deviation of 347 pounds.

a. Find the probability that a randomly selected pumpkin weighs at least 1622 pounds.

b. Find the probability that a randomly selected pumpkin weighs between 465.1 and 1622 pounds.

101.The average annual percentage rate (APR) for credit cards held by U.S. consumers is approximately 15 percent (“Ouch – Credit Card APR Now Tops 15 Percent,” *Time*, January 3, 2012). Suppose the APR for new credit card offers is normally distributed with a mean of 15% and a standard deviation of 4%. What APR must a credit card charge to be in the bottom 10% of all cards?

102.The average annual inflation rate in the United States over the past 98 years is 3.37% and has a standard deviation of approximately 5% (*Inflationdata.com*). In 1980, the inflation rate was above 13%. If the annual inflation rate is normally distributed, what is the probability that inflation will be above 13% next year?

103.The Japan Sumo Association has begun to measure the body fat of wrestlers to try to combat the growing

problem of excessive obesity within the sport. As of 2010, the average wrestler weighed 412 pounds. Suppose the weights of sumo wrestlers are normally distributed, with a standard deviation of 37 pounds. What is the probability that a randomly selected wrestler weighs between 350 and 450 pounds?

lOMoARcPSD|3013804

104.A producer has a history of making bad movies. The movies he has produced have averaged $160,000 dollars at the box office with a standard deviation of $185,000. He thinks his latest movie will be a huge hit. How much will the movie earn at the box office if it is only expected to earn that much or more 0.5% of the time? (Assume that the profit at the box office is normally distributed.)

105.Suppose the amount of time customers must wait to check bags at the ticketing counter in Boston Logan Airport is exponentially distributed with a mean of 14 minutes. What is the probability that a randomly selected customer will have to wait more than 20 minutes?

106.The average wait time to see a doctor at a maternity ward is 16 minutes. What is the probability that a patient will have to wait between 20 and 30 minutes before seeing a doctor? Suppose the wait time is exponentially distributed.

107.The average wait time at a McDonald’s drive-through window is about three minutes (“The Doctor Will See You Eventually,” *The Wall Street Journal*, October 18, 2010). Suppose the wait time is exponentially distributed. What is the probability that a randomly selected customer will have to wait no more than five minutes?

108.Jennifer is waiting for a taxi cab. The average wait time for a taxi is six minutes. Suppose the wait time is exponentially distributed. What is the probability that a taxi arrives in three minutes or less?

lOMoARcPSD|3013804

109.Customers arrive at a drive-through teller window of a bank. They stay in line when the teller is busy.

The service time is exponentially distributed with a mean of four minutes.

a. What is the probability that the next customer in line will take longer than seven minutes to be served? b. What is the probability that the next customer in line will take less than eight minutes to be served? c. What is the probability that the next customer in line will take between three and six minutes to be served?

110.Compute the mean and variance of a lognormal variable *Y* if the mean and the variance of the underlying

normal variable are .

111.After a heavy snow, the city of Boston spends millions of dollars plowing the streets. Suppose the amount of time the city must spend before the streets are clear follows a log-normal distribution. Further suppose that the average amount of time is 12 hours and the standard deviation is 5 hours. What percentage of the time does it take more than 10 hours for the streets to be cleared?

112.Let the household income of residents of the United States be represented by *Y* = e*X*, where *X* is normally distributed. In 2006, the mean U.S. household income was approximately $50,000. Suppose the standard deviation was 16,000. Estimate the proportion of U.S. households that have an income less than 80,000.

113.Compute the mean and variance of a lognormal variable *Y* if the mean and the variance of the underlying

normal variable are .

lOMoARcPSD|3013804

114.The mean household income of France is approximately 20,000 euros. Suppose the household income in France has a standard deviation of 10,000 euros and follows a log-normal distribution. Estimate the proportion of French households that have an income of more than 25,000 euros.

115.Let house prices in a rich community in Chicago be represented by , where *X* is normally distributed. Suppose the mean house price is $1.8 million and the standard deviation is $0.4 million. What is the proportion of the houses that are worth more than $2.5 million?