Problem 1 An electrical system consists of five components. The system works if all of these conditions hold:
component A works,
either of the components B or C work,
component D works, and
component E works.
For components A-E, A-E, let the letter (such as “A”) represent the event that the component works. Assume that the components work independently. Below is a diagram which shows the system components and their reliabilities.
(a) What is the probability that the entire system works? P(BC)=P(B)+P(C) - P( =0.8+0.8-(0.8*0.8) =1.6-0.64
=0.96 P(system works)=P( ( ( ( ( ( P(system works)=0.9*0.96*0.9*0.9 P(system works)=0.7
(b) What is the probability that component Bdoes not work, given that the entire system works?
P(system works)=P( ( ( ( ( P(system works)=0.9*0.8*0.9*0.9 P(system works)=0.58 (c) What is the probability that components B and C both work, given that the entire system works? (a) What is the probability that the entire system works? P(BC)=P(B)+P(C) - P( =0.8+0.8-(0.8*0.8) =1.6-0.64 =0.96 P(system works)=P( ( ( ( ( ( P(system works)=0.9*0.96*0.9*0.9
P(system works)=0.7
Problem 2 A statistics class for engineers consists of 8industrial, 10 mechanical, 10 electrical and 25civil engineering students. As an alternative to a 10 page written report, the professor offers the class the option of a 15 minute oral presentation, an d is surprised to find that every student wants to make the oral presentation. Unfortunately, there is only time for 4 so she decides to randomly select 4 students from this class for an oral presentation.
(a) How many different ways can she choose 4 students from this class? Here n=8+10+10+25=53students r=4 are selected ways =
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(b) How many different ways can she choose a student from each of the four engineering fields?
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(c) What is the probability that she will choose one from each field?
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(d) She is hoping that she won’t pick all four from the same field. What is the probability that she will end up with 4 civil engineering students?
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(d) She wants to make sure that there are 2 civil engineering students. How many different ways can she randomly pick 4 and end up with 2 that are from CE?
( (e) Six industrial engineering students came forward and told the professor that they did not want to be considered for the oral presentation after all. What is the probability that the professor will randomly select 4 students and 2 will be from industrial engineering?
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(g) Continuing from (f), what is the probability that she will select 1 from ME, 1 from EE and 2 from CE?
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Problem 3 A population of 600 semiconductor wafers contains wafers from three lots. The wafers are categorized by lot and by whether they conform to a thickness specification. The following table presents the number of wafers in each category.
For the problems that follow, consider the following two events: A= event that a randomly selected wafer from this population is from Lot A N= event that a randomly selected wafer from this population is Nonconforming in thickness For full credit you must express the requested probabilities in terms of these events and compute the value. (a) What is the probability that it is conforming in thickness?
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( (b) What is the probability that it is from Lot A and is Nonconforming?
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( (b) What is the probability that it is from Lots B or C an d is Conforming?
( ( ( (
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( ( (d) What is the probability that it is from Lots B or C o r Nonconforming?
( ( ( (
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( ( (f) If the wafer is from Lot A, what is the probability that it is conforming?
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( (g) If the wafer is conforming, what is the probability that it is from Lot A?
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(h) If the wafer is conforming, what is the probability that it is not from Lot A?
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( (h) Are A and N independent events? Explain An events those are not affected by previous events are called independents events so A and N are independent events. Problem 4 A researcher for an automobile safety institute was interested in determining whether or n ot the distance that it takes to stop a car going 60 miles per hour depends on the brand of the tire. The researcher measured the stopping distance (in feet) of ten randoml y selected cars for each of five different brands. So that he and his assistants would remain blinded, the researcher arbitrarily labeled the brands of the tires as Brand1, Brand2, Brand3, Brand4, and Brand5. The figure below summarizes their findings.
(a) What type of figure is this, and what five statistics does it show? It appears that the box plots for Brand1 and Brand5 have very little, if any, overlap at all. The same can be said for Brand3 and Brand5. Here are some summary statistics of the data Brand
N
Mean
SD
1
10
188.20
3.88
2
10
195.20
9.02
3
10
187.40
5.27
4
10
191.20
5.55
5
10
200.50
5.44
(b) For each brand, provide the value of a measure of central tendency, and show them from smallest to largest.
It appears that the sample means differ quite a bit. For example, the average stopping distance of Brand3 is 187.4 feet (with a standard deviation of 5.27 feet), while the average stopping distance of Brand5 is 200.5 feet (with a standard deviation of 5.44 feet). A difference of 13 feet could mean the difference between getting into an accident or not. But, of course, we can't draw conclusions about the performance of the brands based on one sample. After all, a different random sample of cars could yield different results. Instead, we need to use the sample means to try to draw conclusions about the population means. (c) Calculate the distance range statisticforeach ofthe 5 brands, and show them from smallest to largest. What type of statistical measure is this? Brand
N
Mean
SD
1
10
188.20
3.88
2
10
195.20
9.02
3
10
187.40
5.27
4
10
191.20
5.55
5
10
200.50
5.44
It appears that the sample means differ quite a bit. For example, the average stopping distance of Brand3 is 187.4 feet (with a standard deviation of 5.27 feet), while the average stopping distance of Brand5 is 200.5 feet (with a standard deviation of 5.44 feet). A difference of 13 feet could mean the difference between getting into an accident or not. But, of course, we can't draw conclusions about the performance of the brands based on one
sample. After all, a different random sample of cars could yield different results. Instead, we need to use the sample means to try to draw conclusions about the population means. (d) Calculate the interquartile distancerange for each of the 5 brands, and show them from smallest to largest. What type of statistical measure is this? In this lesson, we are going to learn how to use a method called analysis of variance to answer the researcher's question. Jumping right to the punch line, with no development or theoretical justification whatsoever, we'll use an analysis of variance table, such as this one: Source
DF
SS
MS
F
P
Brand
4
1174.8
293.7
7.25
0.00
Error
45
1661.7
36.9
Total
49
2836.5
(e) What do measures of dispersion say about stopping distance for the 5 tire brands? To draw conclusions about the equality of two or more population means. And, as we always do when performing hypothesis tests, we'll compare the P-value to α, our desired willingness to commit a Type I error. In this case, the researcher's P-value is very small (0.000, to three decimal places), so he should reject his null hypothesis. That is, there is sufficient evidence, at even a 0.01 level, to conclude that the mean stopping distance for at least one brand of tire is different than the mean stopping distances of the others. So far, we have seen a typical null and alternative hypothesis in the analysis of variance framework, as well as an analysis of variance table. Let's take a look at another example with the idea of continuing to work on developing the basic idea behind the analysis of variance method.
