Problem 1: An electrical system consists of five components. The system works if all of these conditions hold:
1. component A works,
2. either of the components B or C work,
3. component D works, and
4. component E works.
For components 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.
For each part below, express the probability of interest in terms of the events A-E and solve.
(a) What is the probability that the entire system works?
(b) What is the probability that component B does not work, given that the entire system works?
(c) What is the probability that components B and C both work, given that the entire system works?
Problem 2: A statistics class for engineers consists of 8 industrial, 10 mechanical, 10 electrical and 25 civil engineering students. As an alternative to a. 10 page written report, the professor offers the class the option of a 15 minute oral presentation, and 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?
(b) How many different ways can she choose a student from each of the four engineering fields?
(c) What is the probability that she will choose one from each field?
(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?
(e) 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?
(f) 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?
(g) Continuing from (f), what is the probability that she will select 1 from ME. 1 from EE and 2 from CE?
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.
|
LOT
|
Conforming
|
Nonconforming
|
|
A
|
88
|
12
|
|
B
|
165
|
35
|
|
C
|
260
|
40
|
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 wafer is chosen at random from the population.
- What is the probability that it is conforming in thickness?
- What is the probability that it is from Lot A and is Nonconforming?
- What is the probability that it is from Lots B or C and is Conforming?
- What is the probability that it is from Lots B or C or Nonconforming?
- If the wafer is from Lot A, what is the probability that it is conforming?
- If the wafer is conforming, what is the probability that it is from Lot A?
- If the wafer is conforming, what is the probability that it is not from Lot A?
Problem 4: A researcher for an automobile safety institute was interested in determining whether or not 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 randomly 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 Brand I , Brandt, Brand3 , Brand4 , and Brands. The figure below summarizes their findings.
(a) What type of figure is this, and what five statistics does it show?
(b) For each brand, provide the value of a measure of central tendency, and show them from smallest to largest.
(c) Calculate the distance range statistic for each of the 5 brands, and show them from smallest to largest. What type of statistical measure is this?
(d) Calculate the interquartile distance range for each of the 5 brands. and show them from smallest to largest. What type of statistical measure is this?
(e) Do the data provide enough evidence to conclude that at least one of the brands is different from the others with respect to stopping distance? Explain. (1) What do measures of dispersion say about stopping distance for the 5 tire brands?
(g) Suppose Brands 1 and 3 are $200 tires and Brands 2.4 and 5 are each $100 or less. If you had to buy 4 tires, which brand would you buy, and why?