Chapter 4S Supplement Reliability
Chapter 4S
Supplement Reliability
True/False
1.An important dimension of reliability relates to "prescribed operating conditions."
Answer: True
Difficulty: Easy
2.One operational definition of reliability is the probability that a product or system will function on any given trial.
Answer: True
Difficulty: Hard
3.If a system is composed of two parts, both of which must operate and each has a probability of .7 of operating, it is more likely than not that the system will fail.
Answer: True
Difficulty: Hard
4.Redundancy in product design refers to backup parts or systems.
Answer: True
Difficulty: Easy
5."Independent components" cannot exist in the same product since a product by definition is a collection of dependent components.
Answer: False
Difficulty: Medium
6.A component in a product can be considered an "independent" component if it has no dependency on any other component in the product functioning or not.
Answer: True
Difficulty: Easy
7.Events can be considered "independent events" if they have no relation to the occurrence or nonoccurrence of each other.
Answer: True
Difficulty: Easy
Multiple Choice
8.The probability that a product will perform its intended function under prescribed conditions is:
A)reliability
B)merchantability
C)functionality
D)fitness for use
E)manufacturability
Answer: A
Difficulty: Easy
9.A system is comprised of two separate units both of which must function in order for the system to perform as intended. The reliability of one is .4, and the reliability of the other is .5. The overall system reliability is:
A).9
B).3
C).2
D).1
E).02
Answer: C
Difficulty: Easy
10.A product comprised of two components whose failure probabilities are .1 each, has a reliability of:
A)1.8
B).9
C).81
D).8
E).2
Answer: C
Difficulty: Medium
Learning Objective: AS
11.An early warning security fence has three major components each must perform in order for the system to perform. Their reliabilities are .4, .3, and .2. The system reliability is equal to:
A).024
B).076
C).10
D).336
E).90
Answer: A
Difficulty: Medium
12.One of the solar instruments on a satellite has three major parts, all of which must operate. Two have reliabilities of .7 and the other has a reliability of .9. In case the system fails, there is an identical backup system that kicks on automatically. The overall reliability of the system, including the backup, is closest to:
A).30
B).40
C).50
D).60
E).70
Answer: E
Difficulty: Hard
13.A garage door opener has three sections, which have reliabilities of .9, .8, and .9. The lowest of these has an identical backup that automatically goes on if the original fails. All three parts must operate to perform its intended function.The system reliability is closest to:
A).40
B).50
C).60
D).70
E).80
Answer: E
Difficulty: Hard
14.A system is composed of two parts; both must operate in order for the system to perform as intended. The parts have reliabilities of .6 and .5. There is an identical backup system connected to the main system by a switch that has a reliability of .9. The probability that the overall system will operate is closest to:
A).30
B).40
C).50
D).60
E).70
Answer: C
Difficulty: Hard
15.The probability that a product will wear out within a given amount of time is often described by the ______distribution.
A)beta
B)binomial
C)negative exponential
D)normal
E)uniform
Answer: D
Difficulty: Medium
Essay
16.A component of a research instrument has a probability of .98 of operating. It has a backup component that has a probability of .95 of operating. The instrument will function if either component operates. Determine the overall reliability of this subsystem under these conditions:
The backup and the main component are connected by a switch that is certain to function should the main component fail.
The switch to the backup has a probability of .99 of operating.
Answer:
.98 + .02(1.00).95 = .999
.98 + .02(.99).95 = .9988
Difficulty: Medium
17.A computer has three main modules that have individual reliabilities of .80, .90, and .90. Because of recent failures, management is now considering adding redundancy. Determine the reliability of the system with backups under these conditions:
Each module has a backup with a reliability equal to its own and a backup switch with a reliability of 1.00.
The backup consists of an identical computer that operates as a whole rather than backing up individual sections. The single switch for the backup computer has a reliability of .96. Answer:
[.80 + .20(.80)] x [.90 + .10(.90)] x [.80 + .20(.80)] = .9124
[.80(.90).80] + {1- [.80(.90).80] x .96[.80(.90).80]} = .8105
Difficulty: Medium
18.A CD burner has a mean time between failures of 80 weeks. Determine these probabilities:
The unit will not last 64 weeks.
The unit will last for at least 88 weeks.
Neither of the above occurs.
Answer:
MTBF = 80 weeks. [Probabilities are from Table 4-2.]
T = 64, T/MTBF = 64/80 = .801 - .4493 = .5507
T = 88, T/MTBF = 88/80 = 1.10.P(at least 88) = .3329
64 < T < 88 is 1 - (.5507 + .3329) = .1164
Difficulty: Medium
19.A cardiac pace maker can operate an average of 15 months between failures. Determine these probabilities:
(A) a failure within 12 months of the previous failure
(B) a failure between 12 and 18 months after the previous failure
Answer:
MTBF = 15 months
(A) T = 12, T/MTBF = .80.From Table 4-2, P(failure before T) = 1- .4493 = .5507.
(B) 12 <T < 18 is P(failure before 18) - P(failure before 12). 18/15 = 1.2, hence [1 - .3012] - [1 - .4493] = .1481.
Difficulty: Medium
20.The average time between failures for a watch battery is 36 months. Determine the following probabilities:
(A) the probability of failure before 36 months
(B) the probability that a system which is composed of two such solenoids will fail within 36 months if both must work for the system to work
Answer:
T = 36 months, MTBF = 36 months, T/MTBF = 1.00
(A) P(T36) = 1 - .3679 = .6321 [From Table 4-2]
(B) P(failure) = 1 - P(both work) = 1 - .3679(.3679) = .8646.
Difficulty: Medium
21.The useful life of an artificial heart valve is normally distributed with a mean of 48 months and a standard deviation of five months. Determine these probabilities:
The valve will wear out within 44 months of its installation.
The valve will last at least 40 months, before wearing out.
Answer:
m = 48 months
sigma a = 5 months
Difficulty: Medium
22.The estimated service life of a halogen headlight is normally distributed with a mean of 500 hours and a standard deviation of 40 hours. Determine the probability of burning out in these intervals:
(A) before its mean life
(B) after 450 hours of operation
(C) between 540 hours and 600 hours
Answer:
(Appendix B, Table B)
(A) z = 0.00 P(z<0.00) = .5000
(C)
z540 = +1.00 P(z<+1.00) = .8413
z600 = +2.50 P(z<+2.50) = .9938
P(+1.00<z<+2.50) = .9938 - .8413 = .1525
Difficulty: Medium
Multiple Choice
The following information refers to 23-27
An undergraduate business student has purchased a laptop computer for use during exams. This laptop is perfectly reliable except for two parts: its microchip, which has a failure rate of one in every twenty hours of operation; and its battery, which has a failure rate of one in every ten hours of operation. Also, on average the battery will wear out in five hours, with a standard deviation of 30 minutes.
23.Assuming that a new battery has just been installed, what is the probability that the battery will perform reliably during a one-hour exam?
A)0.05
B)0.1
C)0.2
D)0.9
E)0.95
Answer: D
Difficulty: Easy
24.Assuming that a new battery has just been installed, what is the probability that the laptop will perform reliably during a one-hour exam?
A)0.85
B)0.855
C)0.9
D)0.95
E)1.85
Answer: B
Difficulty: Medium
25.Assuming that a new battery has just been installed and the student brings one spare, fully charged battery with him, what is the probability that the laptop will perform reliably during a one-hour exam?
A)0.89775
B)0.9450
C)0.987525
D)0.99
E)0.9975
Answer: B
Difficulty: Medium
26.What is the probability that a new battery will wear out within six hours?
A)0.0228
B)0.4772
C)0.5
D)0.9544
E)0.9772
Answer: E
Difficulty: Medium
27.What maximum battery life should the manufacturer specify in order to have a .8413 probability that a battery will not wear out within that amount of time?
A)3.5 hours
B)4 hours
C)4.5 hours
D)5 hours
E)5.5 hours
Answer: C
Difficulty: Medium
The following information refers to 28-32
The plant manager for the local electric utility company wants to know the reliability of one of his small hydroelectric power generating systems. This system is comprised of two components: a generator which has a random failure rate of one in every two years of operation; and a transformer which has a random failure rate of one in every five years of operation.
28.What is the probability that this system will perform reliably for a period of one year?
A)0.4
B)0.5
C)0.8
D)0.9
E)1.3
Answer: A
Difficulty: Medium
29.If he were to add just a backup generator, what would be the probability that this system would perform reliably for a period of one year?
A)0.48
B)0.60
C)0.72
D)0.75
E)0.96
Answer: B
Difficulty: Medium
30.If he were to add both a backup generator and a backup transformer, what would be the probability that this system would perform reliably for a period of one year?
A)0.48
B)0.60
C)0.72
D)0.75
E)0.96
Answer: C
Difficulty: Medium
31.If the transformer has a mean time to wear-out of 20 years with a standard deviation of 2 years, what is the probability that it will wear out within 19 years?
A)0.1915
B)0.3085
C)0.383
D)0.5
E)0.6915
Answer: B
Difficulty: Medium
32.If the transformer has a mean time to wear-out of 20 years with a standard deviation of 2 years, what service life should the manufacturer specify in order to have a .9332 probability that it will last at least that long before wearing out?
A)17 years
B)18 years
C)19 years
D)20 years
E)21 years
Answer: A
Difficulty: Medium
The following information refers to 33-37
The Ready Light Company produces a flashlight which is perfectly reliable except for two components: the battery, which has a random failure rate of one in every five hours of operation; and the light bulb, which has a random failure rate of three in every ten hours.
33.What is the probability that the battery will perform reliably for one hour?
A)0.2
B)0.3
C)0.5
D)0.7
E)0.8
Answer: E
Difficulty: Easy
34.What is the probability that the flashlight will perform reliably for one hour?
A)0.50
B)0.56
C)0.70
D)0.80
E)1.50
Answer: B
Difficulty: Medium
35.If the company includes a backup light bulb with each flashlight, what is the probability that the flashlight will perform reliably for one hour (excluding light bulb replacement time, if any)?
A)0.6720
B)0.7280
C)0.8736
D)0.9100
E)0.9600
Answer: B
Difficulty: Medium
36.If the company includes both a backup light bulb and a backup battery with each flashlight, what is the probability that the flashlight will perform reliably for one hour (excluding replacement time)?
A)0.6720
B)0.7280
C)0.8736
D)0.9100
E)0.9600
Answer: C
Difficulty: Medium
37.Devices, Inc. supplies Ready Light with the on-off switch for this flashlight. These switches, on average, wear out in fifty hours with a standard deviation of four hours. What service life should Devices specify to have a .8944 probability that a switch will last at least that long before wearing out?
A)43 hours
B)45 hours
C)46 hours
D)50 hours
E)54 hours
Answer: B
Difficulty: Medium
The following information refers to 38-42
The chief of the design team for a new missile wants to know the reliability of its guidance system. This system is comprised of three components: a gyroscope, which has a random failure rate of 3 in every 10 launches; a computer, which has a random failure rate of 2 in every 10 launches; and a rocket motor, which has a random failure of 1 in every 10 launches.
38.What is the probability that the gyroscope will perform reliably during a launch?
A)0.3
B)0.504
C)0.7
D)0.889
E)0.91
Answer: C
Difficulty: Easy
39.What is the probability that the guidance system will perform reliably during a launch?
A)0.3
B)0.504
C)0.7
D)0.889
E)0.91
Answer: B
Difficulty: Medium
40.If they were to connect an identical, backup gyroscope with a perfectly reliable switch to the primary gyroscope, what would be the reliability of the gyroscope function during a launch?
A)0.3
B)0.504
C)0.7
D)0.889
E)0.91
Answer: E
Difficulty: Medium
41.If they were to connect an identical, backup gyroscope with a .90 reliable switch to the primary gyroscope, what would be the reliability of the gyroscope function during a launch?
A)0.3
B)0.504
C)0.7
D)0.889
E)0.91
Answer: D
Difficulty: Medium
42.If the battery used to power the computer has a mean time to wear out in 60 minutes with a standard deviation of 20 minutes, what is the probability that it will last for a launch of 30 minutes duration without wearing out?
A)0.9332
B)0.8664
C)0.5668
D)0.4332
E)0.0668
Answer: A
Difficulty: Medium
The following information refers to 43-47
The chief design engineer for Colonial Warning Systems wants to know the reliability of a two-lamp device intended to warn the operator of potential problems with a commercial steam kettle. One (either) lamp is to be lit if steam pressure drops below the desired level and two (both) are to be lit if temperature drops below a desired level. His supplier estimates that a lamp has a failure rate of .2 per hour of operation, and a mean wear-out time of seven years with a standard deviation of eight months.
43.What is the probability that a lamp will perform reliably for one hour?
A)0
B).2
C).8
D).9
E)1
Answer: C
Difficulty: Easy
44.If the temperature drops below the desired level, what is the probability the system will perform reliably for one hour?
A).04
B).4
C).6
D).64
E).96
Answer: D
Difficulty: Medium
45.If the pressure drops below the desired level, what is the probability the system will perform reliably for one hour?
A).04
B).4
C).6
D).64
E).96
Answer: E
Difficulty: Medium
46.What is the probability that a lamp will last at least until the expiration of the warranty, which is six years?
A).9772
B).9332
C).8664
D).8413
E).4332
Answer: B
Difficulty: Medium
47.What warranty period for a lamp would give a probability of .9987 that it would not require warranty service?
A)5 years
B)5.67 years
C)6.33 years
D)7 years
E)9 years
Answer: A
Difficulty: Medium
Essay
The following information refers to 48-56
A certain product is comprised of two components: X and Y. Component X has a random failure rate of one in every ten years, while component Y's random failure rate is one in every five years. This product has a mean time to wear-out of eight years with a standard deviation of one year.
48.What is the probability that component Y will fail during a year of operation?
Answer: .2
Difficulty: Medium
49.What is the probability that component X will perform reliably for a year?
Answer: .9
Difficulty: Medium
50.What is the probability that component Y will perform reliably for a year?
Answer: .8
Difficulty: Medium
51.What is the probability that this product will perform reliably for a year?
Answer: .72
Difficulty: Medium
52.What would be the reliability of this product if component X were backed up with an identical component?
Answer: .792
Difficulty: Medium
53.What would be the reliability of this product if component Y were backed up with an identical component?
Answer: .864
Difficulty: Medium
54.What would be the reliability of this product if both components were redundant?
Answer: .9504
Difficulty: Medium
55.What is the probability that this product will wear out within eight years? Nine years? Six years?
Answer: .5; 0.8413; .0228
Difficulty: Medium
56.What is the service life for this product which will provide a .5 probability of not wearing out within that amount of time?
A .6915 probability?
A .8413 probability?
A .9332 probability?
Answer: 8 years; 7.5 years; 7 years; 6.5 years
Difficulty: Medium
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