# Your firm manufactures computers. You are sent avery large shipment

August 30, 2017

Question
Your firm manufactures computers. You are sent avery large shipment of chips by a supplier. You want to accept this shipmentonly iften percent or less of the chips are defective, which is the supplier’s claim. Your procedure for deciding whether to accept the shipment is to randomly select a sample of 10 chips, test them, and accept the shipment if there are eitherzero defectivesorone defectiveamong the ten.

1. If in fact ten percent of the shipment is defective, what is the probability that you will accept the shipment?

a. 0.349 c. 0.039

b 0.387 d. 0.736

2. Suppose you discover three defectives among the ten in your sample. Which is “more likely,”(i)there really areten percent defectives, and this particular sample of ten is unusual, or (ii) the true proportion of defectives istwenty percent?

(COMMENT – Conceptually, there are two ways of approaching this question. See the hint below. Both are valid. The calculations are different, because one approach relies on a subjective “prior” probability. In this problem, the conclusions of the two approaches are the same, i.e., the correct answer does not depend on which approach you use.)

a.p = 0.10is more likely b.p = 0.20is more likely 2

3. Supposeinsteadthat the shipment issmall—say your company receives a shipment of twenty chips. Your procedure for deciding to accept the shipment is the same—you randomly select 10 chips, and accept the shipment if, in this sample, you find either zero or one defective. If in fact there are two defective chips in the shipment, what is the probability that you will accept the shipment?

a. 0.763 c. 0.789 b 0.5 d. 0.526

HINT: There are two different ways of approaching the second question. One is to specify a “prior” probability that the defective rate is ten percent, and then use Bayes rule. To implement this approach, assume that the only two possible defective rates are ten percent and twenty percent. Also, assume we want to beagnosticabout the true defective rate—i.e., the prior probability that the defective rate is ten percent is 50%, and the prior probability that the defective rate is twenty percent is 50%. This is the subjectivist orBayesianapproach.

The second approach, which doesn’t address the questionexactly, is to (i) compute the probability of getting three defectives when the defective rate is ten percent, (ii) compute it for a defective rate of twenty percent, and (iii) choose whichever defective rate yields a higher probability. This is in the spirit of thefrequentistapproach.

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