A company is interested in evaluating its current inspection...

A company is interested in evaluating its current inspection procedure for shipments of 50 identical items. The procedure is to take a sample of 5 and pass the shipment if no more than 2 are found to be defective. What proportion of shipments with $20 \%$ defectives will be accepted?

Key Concepts

Hypergeometric Distribution

The hypergeometric distribution is used to model the probability of drawing a certain number of defective items from a finite population when sampling is done without replacement. In situations where the population size is small compared to the sample size, this approach accurately accounts for the decreasing population size as items are drawn, helping to compute the probability that a shipment with a known number of defectives will pass the quality test.

Combinatorial Methods in Probability

Combinatorial methods in probability involve using combinations to count the number of ways events can occur without considering order. These techniques are essential in calculating probabilities in sampling scenarios, such as determining the number of ways to choose defective and non-defective items from a shipment, which is crucial for evaluating the likelihood of accepting a shipment based on a sample test.

Acceptance Sampling

Acceptance sampling is a statistical quality control method used to decide whether to accept or reject a batch of products based on a sample taken from the lot. Instead of inspecting every item, a predetermined number of items are evaluated, and the decision is based on the number of defects found in the sample. This method reduces inspection costs while still providing a level of confidence in the quality of the shipment.

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