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. 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.
1. To have a chance of at most 1% of discarding a lot given that the lot is good, the test should reject if the number of defectives in the sample of size 100 is greater than or equal to _____? (I know this question has a hypergeometric distribution with N=1000, n=100, and the # of Good in population=993. Would the answer be 3 because when x=97 there is a 99.739% chance of not rejecting the lot given that it is good and when x=98 there is a 97.479% chance meaning 3 is the greatest # that has less than or equal to 1% chance of rejection. Is this right?)
2. (continues previous problem) In that case, the chance of rejecting the lot if it really has 70 defective chips is _______? (Would the answer be 6.436% because if N=1000 n=100 and the # of good in population=930 there is a 6.436% probability that x is greater than or equal to 97)
3. (continues previous problem) In the long run, the fraction of lots with 7 defectives that will get discarded erroneously by this test is ________? (For this question I thought you would set N=1000 n=100 # of good in population=993 and find the probability that x is greater than or equal to 97 which is 99.739% and subtract it from 100% which equals .261%. Is this correct?)
4. (continues previous problem) The smallest number of defectives in the lot for which this test has at least a 99% chance of correctly detecting that the lot was bad is ______? (I'm unsure of how to approach this one)
(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 98%, 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 70 defective chips.
5. The number of lots the manufacturer has to produce to get one good lot that is not rejected by the test has a ______? distribution (I thought geometric, other options are negative binomial, normal, hypergeometric, binomial and none of the above) with parameters _________?
6. The expected # of lots the manufacturer must make to get one good lot that is not rejected by the test is ________?
7. 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 ________?