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I have a total population of X, and I test Y < X units for a certainly binary reliability test. Of Y units, some y fail the reliability test.

Is there a method (oversimplification is ok) that can tell me 1) what would be the expected failure rate (or range of failure rate) of the total population X, and 2) what is the expected failure rate (range) if X goes to infinity.

For example, let's assume I have a population of 1000 units, and I test 100 of which 10 fail. The nominal failure rate is 10%, but given the sample size, the likely failure rate is between 2% and 19% (as an example). How can I calculate this range based on the information I have?

useyourhoosh
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  • The obvious answer is the correct one: the expectation of $y/Y$ equals the failure rate in the population. The range you cite is a confidence interval. When the sample size is small compared to the population (yours is right at the threshold), you can use a relatively simple Binomial confidence interval. Otherwise use a Hypergeometric interval, often computed using a "finite population correction." – whuber Jun 09 '23 at 18:02

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