part of the population is excluded from sampling, now what?

[madnak]

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Is it reasonable to "pretend" the sample came from the entire population (the one that has (1 + x)*N elements) and proced as normal?

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Yes. If "There is no particular reason to think the newly found portion of the population is distributed any differently than the original N elements" then taking a sample from N and taking a sample from (N+x) is the same. As we expect the distributions of N and (N+x) to be the same, we expect the distribution of the sample to be the same as well.