Chapter 06 For a continuous random variable x, the probability density function

CHAPTER SIX

CONTINUOUS PROBABILITY DISTRIBUTIONS

MULTIPLE-CHOICE QUESTIONS

In the following multiple-choice questions, circle the correct answer.

1. If arrivals follow a Poisson probability distribution, the time between successive arrivals

must follow

a. a Poisson probability distribution

b. a normal probability distribution

c. a uniform probability distribution

d. an exponential probability distribution

2. Whenever the probability is proportional to the length of the interval in which the random

variable can assume a value, the random variable is

a. uniformly distributed

b. normally distributed

c. exponentially distributed

d. Poisson distributed

3. There is a lower limit but no upper limit for a random variable that follows the

a. uniform probability distribution

b. normal probability distribution

c. exponential probability distribution

d. binomial probability distribution

4. The form of the continuous uniform probability distribution is

a. triangular

b. rectangular

c. bell-shaped

d. a series of vertical lines

5. The mean, median, and mode have the same value for which of the following probability

distributions?

a. uniform

b. normal

c. exponential

d. Poisson

6. The probability distribution that can be described by just one parameter is the

a. uniform

b. normal

c. exponential

d. binomial

7. A continuous random variable may assume

a. all values in an interval or collection of intervals

b. only integer values in an interval or collection of intervals

c. only fractional values in an interval or collection of intervals

d. all the positive integer values in an interval

8. For a continuous random variable x, the probability density function f(x) represents

a. the probability at a given value of x

b. the area under the curve at x

c. Both the probability at a given value of x and the area under the curve at x are

correct answers.

d. the height of the function at x

9. The function that defines the probability distribution of any continuous random variable

is a

a. normal function

b. uniform function

c. Both the normal function and the uniform function are correct.

d. probability density function

10. For any continuous random variable, the probability that the random variable takes on

exactly a specific value is

a. 1.00

b. 0.50

c. any value between 0 to 1

d. zero

11. The uniform probability distribution is used with

a. a continuous random variable

b. a discrete random variable

c. a normally distributed random variable

d. any random variable

12. A uniform probability distribution is a continuous probability distribution where the

probability that the random variable assumes a value in any interval of equal length is

a. different for each interval

b. the same for each interval

c. Either a or b could be correct depending on the magnitude of the standard

deviation.

d. None of the alternative answers is correct.

13. For a uniform probability density function, the height of the function

a. can not be larger than one

b. is the same for each value of x

c. is different for various values of x

d. decreases as x increases

14. A continuous random variable is uniformly distributed between a and b. The probability

density function between a and b is

a. zero

b. (a – b)

c. (b – a)

d. 1/(b – a)

15. The probability density function for a uniform distribution ranging between 2 and 6 is

a. 4

b. undefined

c. any positive value

d. 0.25

16. The random variable x is known to be uniformly distributed between 70 and 90. The

probability of x having a value between 80 and 95 is

a. 0.75

b. 0.5

c. 0.05

d. 1

17. The assembly time for a product is uniformly distributed between 6 to 10 minutes. The

probability density function has what value in the interval between 6 and 10?

a. 0.25

b. 4.00

c. 5.00

d. zero

18. The assembly time for a product is uniformly distributed between 6 to 10 minutes. The

probability of assembling the product between 7 to 9 minutes is

a. zero

b. 0.50

c. 0.20

d. 1

19. The assembly time for a product is uniformly distributed between 6 to 10 minutes. The

probability of assembling the product in less than 6 minutes is

a. zero

b. 0.50

c. 0.15

d. 1

20. The assembly time for a product is uniformly distributed between 6 to 10 minutes. The

probability of assembling the product in 7 minutes or more is

a. 0.25

b. 0.75

c. zero

d. 1

21. The assembly time for a product is uniformly distributed between 6 to 10 minutes. The

expected assembly time (in minutes) is

a. 16

b. 2

c. 8

d. 4

22. The assembly time for a product is uniformly distributed between 6 to 10 minutes. The

standard deviation of assembly time (in minutes) is approximately

a. 0.3333

b. 0.1334

c. 16

d. None of the alternative answers is correct.

Exhibit 6-1

Consider the continuous random variable x, which has a uniform distribution over the interval

from 20 to 28.

23. Refer to Exhibit 6-3. The probability density function has what value in the interval

between 20 and 28?

a. 0

b. 0.050

c. 0.125

d. 1.000

24. Refer to Exhibit 6-1. The probability that x will take on a value between 21 and 25 is

a. 0.125

b. 0.250

c. 0.500

d. 1.000

25. Refer to Exhibit 6-1. The probability that x will take on a value of at least 26 is

a. 0.000

b. 0.125

c. 0.250

d. 1.000

26. Refer to Exhibit 6-1. The mean of x is

a. 0.000

b. 0.125

c. 23

d. 24

27. Refer to Exhibit 6-1. The variance of x is approximately

a. 2.309

b. 5.333

c. 32

d. 0.667

Exhibit 6-2

The travel time for a college student traveling between her home and her college is uniformly

distributed between 40 and 90 minutes.

28. Refer to Exhibit 6-1. What is the random variable in this experiment?

a. the uniform distribution

b. 40 minutes

c. 90 minutes

d. the travel time

29. Refer to Exhibit 6-2. The probability that she will finish her trip in 80 minutes or less is

a. 0.02

b. 0.8

c. 0.2

d. 1.00

30. Refer to Exhibit 6-2. The probability that her trip will take longer than 60 minutes is

a. 1.00

b. 0.40

c. 0.02

d. 0.600

31. Refer to Exhibit 6-2. The probability that her trip will take exactly 50 minutes is

a. zero

b. 0.02

c. 0.06

d. 0.20

32. A normal probability distribution

a. is a continuous probability distribution

b. is a discrete probability distribution

c. can be either continuous or discrete

d. always has a standard deviation of 1

33. Which of the following is not a characteristic of the normal probability distribution?

a. The mean, median, and the mode are equal

b. The mean of the distribution can be negative, zero, or positive

c. The distribution is symmetrical

d. The standard deviation must be 1

34. Which of the following is not a characteristic of the normal probability distribution?

a. The graph of the curve is the shape of a rectangle

b. The total area under the curve is always equal to 1.

c. 99.72% of the time the random variable assumes a value within plus or minus

three standard deviations of its mean

d. The mean is equal to the median, which is also equal to the mode.

35. The highest point of a normal curve occurs at

a. one standard deviation to the right of the mean

b. two standard deviations to the right of the mean

c. approximately three standard deviations to the right of the mean

d. the mean

36. If the mean of a normal distribution is negative,

a. the standard deviation must also be negative

b. the variance must also be negative

c. a mistake has been made in the computations, because the mean of a normal

distribution can not be negative

d. None of the alternative answers is correct.

37. Larger values of the standard deviation result in a normal curve that is

a. shifted to the right

b. shifted to the left

c. narrower and more peaked

d. wider and flatter

38. A standard normal distribution is a normal distribution with

a. a mean of 1 and a standard deviation of 0

b. a mean of 0 and a standard deviation of 1

c. any mean and a standard deviation of 1

d. any mean and any standard deviation

39. In a standard normal distribution, the range of values of z is from

a. minus infinity to infinity

b. -1 to 1

c. 0 to 1

d. -3.09 to 3.09

40. For a standard normal distribution, a negative value of z indicates

a. a mistake has been made in computations, because z is always positive

b. the area corresponding to the z is negative

c. the z is to the left of the mean

d. the z is to the right of the mean

41. For the standard normal probability distribution, the area to the left of the mean is

a. -0.5

b. 0.5

c. any value between 0 to 1

d. 1

42. The mean of a standard normal probability distribution

a. is always equal to 1

b. can be any value as long as it is positive

c. can be any value

d. None of the alternative answers is correct.

43. The standard deviation of a standard normal distribution

a. is always equal to zero

b. is always equal to one

c. can be any positive value

d. can be any value

44. For a standard normal distribution, the probability of z  0 is

a. zero

b. -0.5

c. 0.5

d. one

45. Z is a standard normal random variable. The P(1.20  z  1.85) equals

a. 0.4678

b. 0.3849

c. 0.8527

d. 0.0829

46. Z is a standard normal random variable. The P(1.05 < z < 2.13) equals

a. 0.8365

b. 0.1303

c. 0.4834

d. None of the alternative answers is correct.

47. Z is a standard normal random variable. The P(1.41 < z < 2.85) equals

a. 0.4772

b. 0.3413

c. 0.8285

d. None of the alternative answers is correct.

48. Z is a standard normal random variable. The P(-1.96  z  -1.4) equals

a. 0.8942

b. 0.0558

c. 0.475

d. 0.4192

49. Z is a standard normal random variable. The P (-1.20  z  1.50) equals

a. 0.0483

b. 0.3849

c. 0.4332

d. 0.8181

50. Z is a standard normal random variable. The P(-1.5 < z < 1.09) equals

a. 0.4322

b. 0.3621

c. 0.7953

d. 0.0711

51. Z is a standard normal random variable. The P(z > 2.11) equals

a. 0.4821

b. 0.9821

c. 0.5

d. 0.0174

52. Given that z is a standard normal random variable, what is the value of z if the area to the

right of z is 0.1112?

a. 0.3888

b. 1.22

c. 2.22

d. 3.22

53. Given that z is a standard normal random variable, what is the value of z if the area to the

right of z is 0.1401?

a. 1.08

b. 0.1401

c. 2.16

d. -1.08

54. Given that z is a standard normal random variable, what is the value of z if the area to the

left of z is 0.9382?

a. 1.8

b. 1.54

c. 2.1

d. 1.77

55. Z is a standard normal random variable. What is the value of z if the area between –z and

z is 0.754?

a. 0.377

b. 0.123

c. 2.16

d. 1.16

56. Z is a standard normal random variable. What is the value of z if the area to the right of z

is 0.9803?

a. -2.06

b. 0.4803

c. 0.0997

d. 3.06

57. For a standard normal distribution, the probability of obtaining a z value between -2.4 to

-2.0 is

a. 0.4000

b. 0.0146

c. 0.0400

d. 0.5000

58. For a standard normal distribution, the probability of obtaining a z value of less than 1.6

is

a. 0.1600

b. 0.0160

c. 0.0016

d. 0.9452

59. For a standard normal distribution, the probability of obtaining a z value between

-1.9 to 1.7 is

a. 0.9267

b. 0.4267

c. 1.4267

d. 0.5000

60. X is a normally distributed random variable with a mean of 8 and a standard deviation of

4. The probability that X is between 1.48 and 15.56 is

a. 0.0222

b. 0.4190

c. 0.5222

d. 0.9190

61. X is a normally distributed random variable with a mean of 5 and a variance of 4. The

probability that X is greater than 10.52 is

a. 0.0029

b. 0.0838

c. 0.4971

d. 0.9971

62. X is a normally distributed random variable with a mean of 12 and a standard deviation of

3. The probability that X equals 19.62 is

a. 0.000

b. 0.0055

c. 0.4945

d. 0.9945

63. X is a normally distributed random variable with a mean of 22 and a standard deviation of

5. The probability that X is less than 9.7 is

a. 0.000

b. 0.4931

c. 0.0069

d. 0.9931

64. The ages of students at a university are normally distributed with a mean of 21. What

percentage of the student body is at least 21 years old?

a. It could be any value, depending on the magnitude of the standard deviation

b. 50%

c. 21%

d. 1.96%

Exhibit 6-3

The weight of football players is normally distributed with a mean of 200 pounds and a standard

deviation of 25 pounds.

65. Refer to Exhibit 6-3. What is the random variable in this experiment?

a. the weight of football players

b. 200 pounds

c. 25 pounds

d. the normal distribution

66. Refer to Exhibit 6-3. The probability of a player weighing more than 241.25 pounds is

a. 0.4505

b. 0.0495

c. 0.9505

d. 0.9010

67. Refer to Exhibit 6-3. The probability of a player weighing less than 250 pounds is

a. 0.4772

b. 0.9772

c. 0.0528

d. 0.5000

68. Refer to Exhibit 6-3. What percent of players weigh between 180 and 220 pounds?

a. 34.13%

b. 68.26%

c. 0.3413%

d. None of the alternative answers is correct.

69. Refer to Exhibit 6-3. What is the minimum weight of the middle 95% of the players?

a. 196

b. 151

c. 249

d. None of the alternative answers is correct.

Exhibit 6-4

The starting salaries of individuals with an MBA degree are normally distributed with a mean of

$40,000 and a standard deviation of $5,000.

70. Refer to Exhibit 6-4. What is the random variable in this experiment?

a. the starting salaries

b. the normal distribution

c. $40,000

d. $5,000

71. Refer to Exhibit 6-4. What is the probability that a randomly selected individual with an

MBA degree will get a starting salary of at least $30,000?

a. 0.4772

b. 0.9772

c. 0.0228

d. 0.5000

72. Refer to Exhibit 6-4. What is the probability that a randomly selected individual with an

MBA degree will get a starting salary of at least $47,500?

a. 0.4332

b. 0.9332

c. 0.0668

d. 0.5000

73. Refer to Exhibit 6-4. What percentage of MBA’s will have starting salaries of $34,000 to

$46,000?

a. 38.49%

b. 38.59%

c. 50%

d. 76.98%

Exhibit 6-5

The weight of items produced by a machine is normally distributed with a mean of 8 ounces and a

standard deviation of 2 ounces.

74. Refer to Exhibit 6-5. What is the random variable in this experiment?

a. the weight of items produced by a machine

b. 8 ounces

c. 2 ounces

d. the normal distribution

75. Refer to Exhibit 6-5. What is the probability that a randomly selected item will weigh

more than 10 ounces?

a. 0.3413

b. 0.8413

c. 0.1587

d. 0.5000

76. Refer to Exhibit 6-5. What is the probability that a randomly selected item will weigh

between 11 and 12 ounces?

a. 0.4772

b. 0.4332

c. 0.9104

d. 0.0440

77. Refer to Exhibit 6-5. What percentage of items will weigh at least 11.7 ounces?

a. 46.78%

b. 96.78%

c. 3.22%

d. 53.22%

78. Refer to Exhibit 6-5. What percentage of items will weigh between 6.4 and 8.9 ounces?

a. 0.1145

b. 0.2881

c. 0.1736

d. 0.4617

79. Refer to Exhibit 6-5. What is the probability that a randomly selected item weighs

exactly 8 ounces?

a. 0.5

b. 1.0

c. 0.3413

d. None of the alternative answers is correct.

Exhibit 6-6

The life expectancy of a particular brand of tire is normally distributed with a mean of 40,000 and

a standard deviation of 5,000 miles.

80. Refer to Exhibit 6-6. What is the random variable in this experiment?

a. the life expectancy of this brand of tire

b. the normal distribution

c. 40,000 miles

d. None of the alternative answers is correct.

81. Refer to Exhibit 6-6. What is the probability that a randomly selected tire will have a life

of at least 30,000 miles?

a. 0.4772

b. 0.9772

c. 0.0228

d. None of the alternative answers is correct.

82. Refer to Exhibit 6-6. What is the probability that a randomly selected tire will have a life

of at least 47,500 miles?

a. 0.4332

b. 0.9332

c. 0.0668

d. None of the alternative answers is correct.

83. Refer to Exhibit 6-6. What percentage of tires will have a life of 34,000 to 46,000 miles?

a. 38.49%

b. 76.98%

c. 50%

d. None of the alternative answers is correct.

84. Refer to Exhibit 6-6. What is the probability that a randomly selected tire will have a life

of exactly 47,500 miles?

a. 0.4332

b. 0.9332

c. 0.0668

d. zero

85. Excel’s NORMSDIST function can be used to compute

a. cumulative probabilities for a standard normal z value

b. the standard normal z value given a cumulative probability

c. cumulative probabilities for a normally distributed x value

d. the normally distributed x value given a cumulative probability

86. Excel’s NORMSINV function can be used to compute

a. cumulative probabilities for a standard normal z value

b. the standard normal z value given a cumulative probability

c. cumulative probabilities for a normally distributed x value

d. the normally distributed x value given a cumulative probability

87. Excel’s NORMDIST function can be used to compute

a. cumulative probabilities for a standard normal z value

b. the standard normal z value given a cumulative probability

c. cumulative probabilities for a normally distributed x value

d. the normally distributed x value given a cumulative probability

88. Excel’s NORMINV function can be used to compute

a. cumulative probabilities for a standard normal z value

b. the standard normal z value given a cumulative probability

c. cumulative probabilities for a normally distributed x value

d. the normally distributed x value given a cumulative probability

89. A continuous probability distribution that is useful in describing the time, or space,

between occurrences of an event is a(n)

a. normal probability distribution

b. uniform probability distribution

c. exponential probability distribution

d. Poisson probability distribution

90. The exponential probability distribution is used with

a. a discrete random variable

b. a continuous random variable

c. any probability distribution with an exponential term

d. an approximation of the binomial probability distribution

91. An exponential probability distribution

a. is a continuous distribution

b. is a discrete distribution

c. can be either continuous or discrete

d. must be normally distributed