9-29. A quality-control inspector is testing a batch of printed circuit boards t
ID: 3291938 • Letter: 9
Question
9-29. A quality-control inspector is testing a batch of printed circuit boards to see whether they are capable of performing in a high temperature environment. He knows that the boards that will survive will pass all five of the tests with probability 98%. They will pass at least four tests with probability 99%, and they always pass at least three. On the other hand, the boards that will not survive sometimes pass the tests as well. In fact, 3% pass all five tests, and another 20% pass exactly four. The rest pass at most three tests. The inspector decides that if a board passes all five tests, he will classify it as “good.” Otherwise, he’ll classify it as “bad.” (a) What does a type I error mean in this context? (b) What is the probability of a type I error? (c) What does a type II error mean here? (d) What is the probability of a type II error? 9-30. In the quality-control example of Exercise 9-29, the manager says that the probability of a type I error is too large and that it must be no larger than 0.01. (a) How does this change the rule for deciding whether a board is “good”? (b) How does this affect the type II error? (c) Do you think this reduction in type I error is justified? Explain briefly.
Explanation / Answer
9.29.
a) Here the null hypothesis is that the PCB survives against the alternate that the PCB 'does not survive'. The test says that the PCB will survice if it is classified as 'good'; or, it will not survive if it is classifies as 'bad'. So the Type I error is the error committed when a PCB which can actually survice is classified as 'bad'.
b) Therefore P(Type I error) = P(The PCB is classified as 'bad' | PCB is survives) = 1 - 0.98 = 0.02.
c) The Type II error is the error committed when a PCB which cannot actually survice is classified as 'good'.
d) Therefore P(Type II error) = P(The PCB is classified as 'good' | PCB does not survives) = 0.03.
9.30.
a) If the P(Type I error) <= 0.01 then less than 1% of the articles should be labelled 'bad' when it can actually survice. So the test has to be cahnged as follows: Label a PCB as 'good' if it passes at least 4 tests. In this test
P(Type I error) = 0.01
b) This increass the P(Type II error) to 0.20.
c) Since the reduction in P(Type I error) by 50% increases the P(Type II error) by almost 600% this reduction is not justified in my opinion.
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