Question

A quality-control engineer is testing a batch of car tiers to see whether they are suitable of performing in a high temperature. He knows that the tiers that will survive will pass all five of the quality tests with probability 98%. They will pass at least four quality tests with probability 99%, and they always pass at least three quality tests. On the other hand, the tires that will not survive sometimes pass the tests as well. Indeed, 3% pass all five tests, and another 20% pass exactly four. The rest pass at most three tests. The inspector decides that if a tire passes all five tests, he will classify it as “good.” Otherwise, he’ll classify it as “bad.”

What does a type IIerror mean here?

Answer 1

If we assume that the tires are bad, then probability of passing all the five tests is just 3%. Thus, when a tire passes all the 'five' tests, there is just 3% chance that it is bad. Hence, we are calling a tire good only if we have sufficient evidence it is good. This promots us to think that probably the null hypothesis is that 'the tire is bad'.

Then, the alternative becomes 'the tire is good'. Then, type II error here is just accepting null, when it is false. That is we are concluding the tire is bad, when in reality it isn't.