Suppose that to have power against a wider variety of alternatives, the experimenter decides to base the test on the maximum number of times any face shows, instead of just the number of times one spot shows. That is, she will roll the die 40 times and calculate

(number of times die lands showing one spot)
(number of times die lands showing two spots)
(number of times die lands showing three spots)
(number of times die lands showing four spots)
(number of times die lands showing five spots)
and (number of times die lands showing six spots).

She will reject the null hypothesis if the largest (maximum) of those 6 random numbers is greater than 14.

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.

1. To have a chance of at most 3% 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 _____.

2. In that case, the chance of rejecting the lot if it really has 50 defective chips is ____.

3. In the long run, the fraction of lots with 7 defectives that will get discarded erroneously by this test is ____.

4. The smallest number of defectives in the lot for which this test has at least a 97% chance of correctly detecting that the lot was bad is _____.