(adsbygoogle = window.adsbygoogle || []).push({}); Problem:

Suppose I have a production process that yields output in batches ofnitems. For each batch, I can test whether they are of good or bad quality. Letq_i ∈ {1,0}be the quality of tested itemi.

If more than half of the items are ‘bad’, the batch should be discarded. In other words: the batch should be discarded if the average quality of items isq(n)<0.5.

Suppose testing is sequential (i.e. you learn the quality of items one at the time), and each test is a random draw from a Bernoulli distribution with unknown meanp.

I would like to know when to stop testing items in a batch, in order to decide whether to discard it without testing all items, and yet be statistically confident that the outcome (discard vs not discard) is the same as the one that would have been reached if all items had been tested.

What I currently have:

If I knewp, I could use a normal approximation to estimate the confidence interval of the binomial proportionq(k)for anyk. With this, for any arbitrary width of the interval, I could calculate the number of tests that are required to achieve a given confidence level (let me call such number of testsk*).

Since I don’t knowp, one option would be to solve fork*assuming the worst case scenario (i.e.p=0.5). This was proposed (and discussed) in a related question here. Another similar alternative was proposed here.

This approach, however, does not take into account thatnis finite, so once many items have been tested, it becomes very unlikely that one more test will swing the outcome...

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# A Stopping rule for quality control problem

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