1. Acceptance Sampling. Suppose a company has purchased a quantity of items that
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Question
1. Acceptance Sampling. Suppose a company has purchased a quantity of items that have been mass-produced. The manufacturing process is not perfect, and some of the items are expected to be defective. The company cannot test every one of the items before purchasing (too expensive!), so instead, they employ acceptance sampling: Take a sample (without replacement) from the shipment and inspect the sample for defects. If the sample is defect-free, they accept the entire shipment. If any of the items in the sample is defective, they reject the shipment.
Suppose an airport orders 810 baggage scales and suppose (unknown to the airport) a lot of the scales, 102 of them, are defective. Using acceptance sampling with a sample size of 3:
a. Is it appropriate to treat this sampling as independent, even though we are sampling without replacement? Explain.
Explanation / Answer
There are some situations where sampling with or without replacement does not substantially change any probabilities of selecting a defective piece. Suppose that we are randomly choosing 2 scales from a shipment of 810 baggage scales in which 102 of them are defective.
If we sample with replacement, then the probability of choosing a defective scale on the first selection is given by 102/810 = 0.126. The probability of a defective scale on the second selection is still 0.126. The probability of both scale being defective is 0.126 x 0.126 = 0.0159.
If we sample without replacement then the first probability is unaffected. The second probability is now 101/809 = 0.125 , which is extremely close to 0.126. The probability that both are defective is 0.126 x 0.125 = 0.0158.
The probabilities are technically different, however they are close enough to be nearly indistinguishable. For this reason, many times even though we sample without replacement, we treat the selection of each item as if they are independent of the other items in the sample.
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