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We have a population of unknown size. Each element has a class A or B. We are trying to estimate the proportion of A's in the population. We can randomly sample from the population and for each element in the sample we can apply a test which will either tell us if the sample is of class A or B but in certain cases the test might fail and not give a response at all.

What is the size of sample required to get the estimate with a given confidence interval? How do we correct for the "No Response" bias? What if the population size was know?

If someone would could point me to the appropriate reading material, that would be helpful. I am looking for detailed proofs that derive such a bound, if one exists.

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One of the joys of population sampling is that the sample size needed for population estimates of a predefined accuracy is invariant to population size. Have a look at the Wikipedia page on sample size determination and you can see the formula for the proportion case as the first example under part 2.

Non-response bias, on the other hand, remains a serious issue in sampling. This term is typically used with surveys of people (who can be absent when sampling occurs or refuse to answer the survey). I am a little confused with your wording which suggests that you're using a test, and the test will not provide a clear-cut value, and I am not familiar with the use of the term "no response" in this situation. Are you doing something like a diagnostic test, and sometimes the method gives an inconclusive result?

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