Question

A manufacturer of computer memory chips produces chips in lots of 1000. If nothing has gone wrong in the manufacturing process, at most 7 chips each lot would be defective, but if something does go wrong, there could be far more defective chips. If something goes wrong with a given lot, they discard the entire lot. It would be prohibitively expensive to test every chip in every lot, so they want to make the decision of whether or not to discard a given lot on the basis of the number of defective chips in a simple random sample. They decide they can afford to test 100 chips from each lot. You are hired as their statistician.

There is a tradeoff between the cost of eroneously discarding a good lot, and the cost of warranty claims if a bad lot is sold. The next few problems refer to this scenario.

(Continues previous problem.) Suppose that whether or not a lot is good is random, that the long-run fraction of lots that are good is 95%, and that whether each lot is good is independent of whether any other lot or lots are good. Assume that the sample drawn from a lot is independent of whether the lot is good or bad. To simplify the problem even more, assume that good lots contain exactly 7 defective chips, and that bad lots contain exactly 20 defective chips.

Problem 15 (Continues previous problem.) The number of lots the manufacturer has to produce to get one good lot that is not rejected by the test has a (Q20) geometric v distribution, with parameters (Q21) ? Problem 16 (Continues previous problem.) The expected number of lots the manufacturer must make to get one good lot that is not rejected by the test is (Q22) 1.0259 Problem 17 (Continues previous problem.) With this test and this mix of good and bad lots, among the lots that pass the test, the long-run fraction of lots that are actually bad is (Q23)|

Answer 1

Problem : 15 )

You're right about geometric distribution.

The parameter of this distribution, theta, is the probability that a good lot isn't rejected by the test.

It is theta = P(Accept H0 | H0 is true) = 1- P(Reject H0 | H0 is true),

And you know that P(Reject H0 | H0 is true) is the type one error of the test.

If, as we take it .01 we have theta=1-.01=.99.

Problem : 16 )

The number of lots the manufacturer must make to get one good lot that is not rejected has a geometric

distribution with parameter theta=.99,

So the expected value of number of lots the manufacturer must make to get one good lot that is not rejected is:

1/theta=1/.99.

= 1.0101.

So the expected value = 1.0101.

Problem : 17)

With this test and this mix of good and bad lots, among the lots that pass the test, the long-run fraction of lots

That are actually bad is,

P(Accept H0 | H1 is true) = P(Reject H1 | H1 is true) = the type two error of the test.

Therefore,

With this test and this mix of good and bad lots, among the lots that pass the test, the long-run fraction of lots that are actually bad is type two error of the test.