1. The volume of liquid in an unopened 1- gallon can of paint is an example of _________.
a) the binomial distribution
b) both discrete and continuous variable c) a continuous random variable
d) a discrete random variable
e) a constant
2. The number of defective parts in a lot of 25 parts is an example of _______.
a) a discrete random variable
b) a continuous random variable c) the Poisson distribution
d) the normal distribution
e) a constant
A market research team compiled the following discrete probability distribution. In this distribution, x represents the number of automobiles owned by a family.
Answer questions 3-5 based on the above discrete probability distribution.
x
P(x)
0 0.10
1 0.10
2 0.50
3 0.30
3. The mean (average) value of x is _____.
a) 1.0 b) 1.5 c) 2.0 d) 2.5 e) 3.0
4. The standard deviation of x is ________.
a) 0.80 b) 0.89 c) 1.00 d) 2.00 e) 2.25
5. Which of the following statements is true?
a) This distribution is skewed to the right. b) This is a binomial distribution.
c) This is a normal distribution.
d) This distribution is skewed to the left. e) This distribution is bimodal.
6. Twenty five items are randomly selected from a batch of 1000 items. Each of these items has the same probability of being defective. The probability that exactly 2 of the 25 are defective could best be found by _______.
a) using the normal distribution
b) using the binomial distribution
c) using the Poisson distribution
d) using the exponential distribution e) using the uniform distribution
7. A fair coin is tossed 5 times. What is the probability that exactly 2 heads are observed?
a) 0.313 b) 0.073 c) 0.400 d) 0.156 e) 0.250
Pinky Bauer, Chief Financial Officer of Harrison Haulers, Inc., suspects irregularities in the payroll system, and orders an inspection of a random sample of vouchers issued since January 1, 2006. A sample of ten vouchers is randomly selected, without replacement, from the population of 2,000 vouchers. Each voucher in the sample is examined for errors and the number of vouchers in the sample with errors is denoted by x.
Answer questions 8-11 based on the above information.
8. If 20% of the population of vouchers contain errors, P(x = 0) is _____________.
a) 0.8171 b) 0.1074 c) 0.8926 d) 0.3020 e) 0.2000
9. If 20% of the population of vouchers contain errors, P(x > 0) is _____________.
a) 0.8171 b) 0.1074 c) 0.8926 d) 0.3020 e) 1.0000
10. If 20% of the population of vouchers contains errors, the mean value of x is ____.
a) 400 b) 2 c) 200 d) 5 e) 1
11. If 20% of the population of vouchers contains errors, the standard deviation of x is ______.
a) 1.26 b) 1.60 c) 14.14 d) 3.16 e) 0.00
12. If x is a binomial random variable with n=8 and p=0.6, what is the probability that x is equal to 4?
a) 0.500 b) 0.005 c) 0.124 d) 0.232 e) 0.578
13. If x is a binomial random variable with n = 12 and p = 0.45, P(4 ≤ x ≤ 6) is _______?
a) 0.1700 b) 0.2225 c) 0.2124 d) 0.5838 e) 0.6048
14. If x is n=10 and
a) 0.6177 b) 0.2508 c) 0.3823 d) 0.6331 e) 0.3669
15. If x is n=20 and
a) 0.0654 b) 0.2277 c) 0.8867 d) 0.1144 e) 0.1133
16. If x is n=20 and
a) 0.0867 b) 0.0432 c) 0.1330 d) 0.8670 e) 0.0898
a binomial random variable with p=0.6, P(x ≥ 6) is _______?
a binomial random variable with p=0.3, P(x > 8) is _______?
a binomial random variable with p=0.9, P(x ≤ 16) is _______?
According to Cerulli Associates of Boston, 30% of all CPA financial advisors have an average client size between $500,000 and $1 million. Thirty-four percent have an average client size between $1 million and $5 million. Suppose a complete list of all CPA financial advisors is available and 18 are randomly selected from that list.
Answer the questions 17-22 based on the above information.
17. What is the expected number of CPA financial advisors that have an average client size between $500,000 and $1 million?
a) 0.30 b) 0.612 c) 6.12 d) 5.40 e) 0.54
18. What is the expected number with an average client size between $1 million and $5 million?
a) 0.34 b) 6.12 c) 0.612 d) 5.40 e) 0.54
19. What is the probability that at least eight CPA financial advisors have an average client size between $500,000 and $1 million?
a) 0.1407 b) 0.0811 c) 0.0596 d) 0.9404 e) 0.8593
20. What is the probability that two, three, or four CPA financial advisors have an average client size between $1 million and $5 million?
a) 0.0229 b) 0.0630 c) 0.1217 d) 0.7924 e) 0.2076
21. What is the probability that none of the CPA financial advisors have an average client size between $500,000 and $1 million?
a) 0.0006 b) 0.9994 c) 0.0016 d) 0.0084 e) 0.0126
22. What is the probability that none have an average client size between $1 million and $5 million?
a) 0.0016 b) 0.9994 c) 0.0084 d) 0.0006 e) 0.0126
23. The number of cars arriving at a toll booth in five-minute intervals is Poisson distributed with a mean of 3 cars arriving in five-minute time intervals. The probability of 5 cars arriving over a five-minute interval is _______.
a) 0.0940 b) 0.0417 c) 0.1500 d) 0.1008 e) 0.2890
24. The number of cars arriving at a toll booth in five-minute intervals is Poisson distributed with a mean of 3 cars arriving in five-minute time intervals. The probability of 3 cars arriving over a five-minute interval is _______.
a) 0.2700 b) 0.0498 c) 0.2240 d) 0.0001 e) 0.0020
25. Suppose that, for every lot of 100 computer chips a company produces, an average of 1.4 are defective. Another company buys many lots of these chips at a time, from which one lot is selected randomly and tested for defects. If the tested lot contains more than three defects, the buyer will reject all the lots sent in that batch. What is the probability that the buyer will accept the lots? Assume that the defects per lot are Poisson distributed.
a) 0.9463 b) 0.0537 c) 0.1128 d) 0.2417 e) 0.3452
A medical researcher estimates that .00004 of the population has a rare blood disorder. If the researcher randomly selects 100,000 people from the population, Answer questions 26-27 based on the above information using Poisson Approximation to Binomial problems.
26. What is the probability that seven or more people will have the rare blood disorder?
a) 0.0298 b) 0.0511 c) 0.8894 d) 0.0595 e) 0.1106
27. What is the probability that more than 10 people will have the rare blood disorder?
a) 0.0081 b) 0.9972 c) 0.0019 d) 0.0028 e) 0.9919
A high percentage of people who fracture or dislocate a bone see a doctor for that condition. Suppose the percentage is 99%. Consider a sample in which 300 people are randomly selected who have fractured or dislocated a bone.
Answer questions 28-30 based on the above information using Poisson Approximation to Binomial problems.
28. What is the expected number of people who would not see a doctor?
a) 297 b) 3 c) 30 d) 300 e) 1
29. What is the probability that exactly five of them did not see a doctor?
a) 0.0504 b) 0.9161 c) 0.1008 d) 0.1680 e) 0.8992
30. What is the probability that fewer than four of them did not see a doctor?
a) 0.1680 b) 0.8153 c) 0.1008 d) 0.2528 e) 0.6472
31. Assume that a random variable has a Poisson distribution with a mean of 5 occurrences per ten minutes. The number of occurrences per hour follows a Poisson distribution with λ equal to _________
a) 5 b) 60 c) 30 d) 10 e) 20
32. The Poisson distribution is being used to approximate a binomial distribution. If n=40 and p=0.06, what value of lambda would be used?
a) 0.06 b) 2.4 c) 0.24 d) 24 e) 40
33. The number of phone calls arriving at a switchboard in a 10 minute time period would best be modeled with the _________.
a) binomial distribution
b) hypergeometric distribution c) Poisson distribution
d) hyperbinomial distribution e) exponential distribution
34. The number of defects per 1,000 feet of extruded plastic pipe is best modeled with the ________________.
a) Poisson distribution
b) Pascal distribution
c) binomial distribution
d) hypergeometric distribution e) exponential distribution
35. The hypergeometric distribution must be used instead of the binomial distribution when ______
a) sampling is done with replacement
b) sampling is done without replacement c) n≥5% N
d) both b and c
e) there are more than two possible outcomes
36. The probability of selecting 3 defective items and 7 good items from a warehouse containing 10 defective and 50 good items would best be modeled with the _______.
a) binomial distribution
b) hypergeometric distribution c) Poisson distribution
d) hyperbinomial distribution e) exponential distribution
Circuit boards for wireless telephones are etched, in an acid bath, in batches of 100 boards. A sample of seven boards is randomly selected from each lot for inspection. A batch contains two defective boards; and x is the number of defective boards in the
sample.
Answer questions 37-39 based on the above information.
37. P(x=1) is _______.
a) 0.1315 b) 0.8642 c) 0.0042 d) 0.6134 e) 0.6789
38. P(x=2) is _______.
a) 0.1315 b) 0.8642 c) 0.0042 d) 0.6134 e) 0.0034
39. P(x=0) is _______.
a) 0.1315 b) 0.8642 c) 0.0042 d) 0.6134 e) 0.8134
40. A large industrial firm allows a discount on any invoice that is paid within 30 days. Of all invoices, 10% receive the discount.
In a company audit, 10 invoices are sampled at random. The probability that fewer than 3 of the 10 sampled invoices receive the discount is approximately __________.
a) 0.1937 b) 0.057 c) 0.001 d) 0.3486 e) 0.9298
41. In a certain communications system, there is an average of 1 transmission error per 10 seconds. Assume that the distribution of transmission errors is Poisson. The probability of 1 error in a period of one-half minute is approximately ________.
a) 0.1493 b) 0.3333 c) 0.3678 d) 0.1336 e) 0.03
42. It is known that screws produced by a certain company will be defective with probability .01 independently of each other. The company sells the screws in packages of 25 and offers a money-back guarantee that at most 1 of the 25 screws is defective. Using Poisson approximation for binomial distribution, the probability that the company must replace a package is approximately _________
a) 0.01 b) 0.1947 c) 0.7788 d) 0.0264 e) 0.2211
On Monday mornings, the First National Bank only has one teller window open for deposits and withdrawals. Experience has shown that the average number of arriving customers in a four-minute interval on Monday mornings is 2.8, and each teller can serve more than that number efficiently. These random arrivals at this bank on Monday mornings are Poisson distributed.
Answer the questions 43-50 based on the above information.
43. What is the probability that on a Monday morning exactly six customers will arrive in a four-minute interval?
a) 0.9756 b) 0.0872 c) 0.9593 d) 0.0163 e) 0.0407
44. What is the probability that no one will arrive at the bank to make a deposit or withdrawal during a four-minute interval?
a) 0.9392 b) 0.1703 c) 0.0608 d) 0.0000 e) 0.8297
45. Suppose the teller can serve no more than four customers in any four-minute interval at this window on a Monday morning. What is the probability that, during any given four-minute interval, the teller will be unable to meet the demand?
a) 0.8477 b) 0.1523 c) 0.1557 d) 0.8443 e) 0.3081
46. Suppose the teller can serve no more than four customers in any four-minute interval at this window on a Monday morning. What is the probability that the teller will be able to meet the demand?
a) 0.8477 b) 0.1557 c) 0.8443 d) 0.1523 e) 0.3081
47. When demand cannot be met during any given interval, a second window is opened. What percentage of the time will a second window have to be opened?
a) 0.8477 b) 0.8443 c) 0.1557 d) 0.1523 e) 0.3081
48. What is the probability that exactly three people will arrive at the bank during a two- minute period on Monday mornings to make a deposit or a withdrawal?
a) 0.1082 b) 0.0026 c) 0.2225 d) 0.1128 e) 0.0407
49. What is the probability that five or more customers will arrive during an eight minute period?
a) 0.1523 b) 0.0143 c) 0.6579 d) 0.3421 e) 0.8477
50. On Saturdays, cars arrive at Sami Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen minute interval. Using the Poisson distribution, the probability that five cars will arrive during the next five minute interval is _____________.
a) 0.1008 b) 0.0361 c) 0.1339 d) 0.1606 e) 0.3610
Chp. 6: Questions 51-100.
51. If x is uniformly distributed over the interval 8 to 12, inclusively (8 x 12), then the height of this distribution, f(x), is ...
a) 1/8 b) 1/4 c) 1/12 d) 1/20 e) 1/24
52. If x is uniformly distributed over the interval 8 to 12, inclusively (8 x 12), then the mean of this distribution is _____.
a) 10
b) 20
c) 5
d) 0
e) unknown
53. If x is uniformly distributed over the interval 8 to 12, inclusively (8 x 12), then the standard deviation of this distribution is __________________.
a) 4.00 b) 1.33 c) 1.15 d) 2.00 e) 1.00
54. If x is uniformly distributed over the interval 8 to 12, inclusively (8 x 12), then the probability, P(9 x 11), is ____.
a) 0.250 b) 0.500 c) 0.333 d) 0.750 e) 1.000
55. If x is uniformly distributed over the interval 8 to 12, inclusively (8 x 12), then the probability, P(10.0 x 11.5), is _.
a) 0.250 b) 0.333 c) 0.375 d) 0.500 e) 0.750
56. If x is uniformly distributed over the interval 8 to 12, inclusively (8 x 12), then the probability, P(13 x 15), is __________________.
a) 0.250 b) 0.500 c) 0.375 d) 0.000 e) 1.000
57. If x is uniformly distributed over the interval 8 to 12, inclusively (8 x 12), then P(x < 7) is __________________.
a) 0.500 b) 0.000 c) 0.375 d) 0.250 e) 1.000
58. If x is uniformly distributed over the interval 8 to 12, inclusively (8 x 12), then P(x 11) is ________.
a) 0.750 b) 0.000 c) 0.333 d) 0.500 e) 1.000
59. If x is uniformly distributed over the interval 8 to 12, inclusively (8 x 12), then P(x 10) is __________________.
a) 0.750 b) 0.000 c) 0.333 d) 0.500 e) 0.900
60. If a continuous random variable x is uniformly distributed over the interval 8 to 12, inclusively, then P(x = exactly 10) is __.
a) 0.750 b) 0.000 c) 0.333 d) 0.500 e) 0.900
61. The normal distribution is an example of
a) a discrete distribution
b) a continuous distribution c) a bimodal distribution
d) an exponential distribution e) a binomial distribution
62. The total area underneath any normal curve is equal to _______.
a) the mean
b) one
c) the variance
d) the coefficient of variation e) the standard deviation
63. The area to the left of the mean in any normal distribution is equal to _______.
a) the mean
b) 1
c) the variance d) 0.5
e) -0.5
64. A standard normal distribution has the following characteristics:
a) the mean and the variance are both equal to 1
b) the mean and the variance are both equal to 0
c) the mean is equal to the variance
d) the mean is equal to 0 and the variance is equal to 1
e) the mean is equal to the standard deviation
65. If x is a normal random variable with mean 80 and standard deviation 5, the zscore for x = 88 is ________.
a) 1.8 b) -1.8 c) 1.6 d) -1.6 e) 8.0
66. Suppose x is a normal random variable with mean 60 and standard deviation 2.
A z score was calculated for a number, and the z score is 3.4. What is x?
a) 63.4
b) 56.6 c) 68.6 d) 53.2 e) 66.8
67. Suppose x is a normal random variable with mean 60 and standard deviation 2.
A z score was calculated for a number, and the z score is -1.3. What is x?
a) 58.7 b) 61.3 c) 62.6 d) 57.4 e) 54.7
68. Let z be a normal random variable with mean 0 and standard deviation 1. What is P(z < 1.3)?
a) 0.4032 b) 0.9032 c) 0.0968 d) 0.3485 e) 0. 5485
69. Let z be a normal random variable with mean 0 and standard deviation 1. What is P(1.3 < z < 2.3)?
a) 0.4032 b) 0.9032 c) 0.4893 d) 0.0861 e) 0.0086
70. Let z be a normal random variable with mean 0 and standard deviation 1. What is P(z > 2.4)?
a) 0.4918 b) 0.9918 c) 0.0082 d) 0.4793 e) 0.0820
71. Let z be mean 0 and P(z < -2.1)?
a) 0.4821 b) -0.4821 c) 0.9821 d) 0.0179 e) -0.0179
72.Let z be a normal random variable with mean 0 standard deviation 1. What isP(z > -1.1)?
a) 0.36432 b) 0.8643 c) 0.1357 d) -0.1357 e) -0.8643
73.Let z be a normal random variable with mean 0 and standard deviation 1. What is P(-2.25 < z < -1.1)?
a) 0.36432 b) 0.8643 c) 0.1357 d) -0.1357 e) -0.8643
74. The expected (mean) life of a particular type of light bulb is 1,000 hours with a standard deviation of 50 hours. The life of this bulb is normally distributed. What is the probability that a randomly selected bulb would last longer than 1150 hours?
a) 0.4987 b) 0.9987 c) 0.0013 d) 0.5013 e) 0.5513
75. The expected (mean) life of a particular type of light bulb is 1,000 hours with a standard deviation of 50 hours. The life of
a) 0.3643 b) 0.8643 c) 0.1235 d) 0.4878 e) 0.5000
76. The expected (mean) life of a particular type of light bulb is 1,000 hours with a standard deviation of 50 hours. The life of this bulb is normally distributed. What is the probability that a randomly selected bulb would last fewer than 940 hours?
a) 0.3849 b) 0.8849 c) 0.1151 d) 0.6151 e) 0.6563
77. Suppose you are working with a data set that is normally distributed with a mean of 400 and a standard deviation of 20. Determine the value of x such that 60% of the values are greater than x.
a) 404.5 b) 395.5 c) 405.0 d) 395.0 e) 415.0
According to a report by Scarborough Research, the average monthly household cellular phone bill is $60. Suppose local monthly household cell phone bills are normally distributed with a standard deviation of $11.35.
Answer questions 78-81 based on the above information.
78. What is the probability that a randomly selected monthly cell phone bill is more than $85?
a) 0.4861 b) 0.9861 c) 0.6139 d) 0.5000 e) 0.0139
79. What is the probability that a randomly selected monthly cell phone bill is between $45 and $70?
a) 0.8106 b) 0.9066 c) 0.7172 d) 0.4066 e) 0.3106
80. What is the probability that a randomly selected monthly cell phone bill is between $65 and $75?
a) 0.2366 b) 0.1700 c) 0.4066 d) 0.0934 e) 0.6700
81. What is the probability that a randomly selected monthly cell phone bill is no more than $40?
a) 0.4987 b) 0.4608 c) 0.5000 d) 0.9608 e) 0.0392
82. According to Student Monitor, a New Jersey research firm, the average cumulated college student loan debt for a graduating senior is $25,760.Assume that the standard deviation of such student loan debt is $5,684. Thirty percent of these graduating seniors owe more than what amount?
a) $28,715.68 b) $2,955.68 c) $22,804.32 d) $28,809.28 e) $28,359.68
83. Let x be a binomial random variable with n=20 and p=.8. If we use the normal distribution to approximate probabilities for this, we would use a mean of _______.
a) 20 b) 16 c) 3.2 d) 8 e) 5
84. Let x be a binomial random variable with n=100 and p=.8. If we use the normal distribution to approximate probabilities for this, a correction for continuity should be made. To find the probability of more than 12 successes, we should find _______.
a) P(x>12.5) b) P(x>12) c) P(x>11.5) d) P(x<11.5) e) P(x < 12)
A study about strategies for competing in the global marketplace states that 52% of the respondents agreed that companies need to make direct investments in foreign countries. It also states that about 70% of those responding agree that it is attractive to have a joint venture to increase global competitiveness. Suppose CEOs of 95 manufacturing companies are randomly contacted about global strategies.
Using Normal Approximation of Binomial Distribution with correction for continuity, answer questions 85-88 based on above information.
85. What is the probability that between 44 and 52 (inclusive) CEOs agree that companies should make direct investments in foreign countries?
a) 0.3869 b) 0.2389 c) 0.6258 d) 0.5013 e) 0.7389
86. What is the probability that more than 56 CEOs agree with that assertion?
a) 0.4279 b) 0.8279 c) 0.5000 d) 0.0721 e) 0.5721
87. What is the probability that fewer than 60 CEOs agree that it is attractive to have a joint venture to increase global competitiveness?
a) 0.5000 b) 0.0582 c) 0.4418 d) 0.9418 e) 0.5582
88. What is the probability that between 55 and 62 (inclusive) CEOs agree with that assertion?
a) 0.4963 b) 0.9963 c) 0.3133 d) 0.8099
e) 0.1830
89. The average length of time between arrivals at a turnpike tollbooth is 23 seconds. Assume that the time between arrivals at the tollbooth is exponentially distributed. What is the probability that a minute or more will elapse between arrivals?
a) 0.9265 b) 0.0435 c) 0.4365 d) 0.0735 e) 0.5000
90. The average length of time between arrivals at a turnpike tollbooth is 23 seconds. Assume that the time between arrivals at the tollbooth is exponentially distributed. If a car has just passed through the tollbooth, what is the probability that no car will show up for at least 3 minutes?
a) 0.0004 b) 0.9996 c) 0.4996 d) 0.0435 e) 0.9265
During the summer at a small private airport in western Nebraska, the unscheduled arrival of airplanes is Poisson distributed with an average arrival rate of 1.12 planes per hour.
Answer questions 91-93 based on the above information.
91. What is the average interarrival time between planes (in minutes)?
a) 53.6 b) 67.2 c) 53.4 d) 60
e) 58.88
92. What is the probability that at least 2 hours will elapse between plane arrivals?
a) 0.5000 b) 0.8935 c) 0.3935 d) 0.6065 e) 0.1065
93. What is the probability of two planes arriving less than 10 minutes apart?
a) 0.8297 b) 0.1703 c) 0.6703 d) 0.3297 e) 0.5000
94. The probability that a call to an emergency help line is answered in less than 10 seconds is 0.8. Assume that the calls are independent of each other. Using the normal approximation for binomial with a correction for continuity, the probability that at least 75 of 100 calls are answered within 10 seconds is approximately _______
a) 0.8
b) 0.1313 c) 0.5235 d) 0.9154 e) 0.8687
95. Inquiries arrive at a record message device according to a Poisson process of rate 15 inquiries per minute. The probability that it takes more than 12 seconds for the first inquiry to arrive is approximately _________
a) 0.05
b) 0.75
c) 0.25
d) 0.27
e) 0.73
96. On Saturdays, cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen minute interval. The probability that at least 2 minutes will elapse between car arrivals is _____________.
a) 0.0000 b) 0.4493 c) 0.1353 d) 1.0000 e) 1.0225
97. On Saturdays, cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen minute interval. The probability that less than 10 minutes will elapse between car arrivals is _________.
a) 0.8465 b) 0.9817 c) 0.0183 d) 0.1535 e) 0.2125
98. Incoming phone calls generally are thought to be Poisson distributed. If an operator averages 2.2 phone calls every 30 seconds, what is the expected (average) amount of time between calls (in seconds)?
a) 66
b) 30
c) 13.64 d) 60
e) 27.27
99. Incoming phone calls generally are thought to be Poisson distributed. If an operator averages 2.2 phone calls every 30 seconds, what is the probability that a minute or more would elapse between incoming calls?
a) 0.9877 b) 0.5123
c) 0.4877 d) 0.5000 e) 0.0123
100. Incoming phone calls generally are thought to be Poisson distributed. If an operator averages 2.2 phone calls every 30 seconds, what is the probability that at least two minutes would elapse between incoming calls?
a) 0.0002 b) 0.9998 c) 0.4998 d) 0.5000 e) 0.5002