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My question is about the validity of a certain technique that I have seen in my workplace. This concerns analysis of a dataset that I am getting from a third-party. The details aren't important to me, I am just interested in whether the technique being used is statistically valid.

So the third party is drawing a sample of customers from a database and using this sample to make conclusions about a total customer base. Assume that the sample is basically representative of the population across the relevant attributes. Straightforward.

However, what if we want to know about a subset of this population? For various reasons, it isn't possible to take a sample for this subset. What do we do?

My first question is whether we can use the original sample to draw conclusions about the subset? Is this valid? My understanding is that you can only do this if your sample is also representative of the subset. Is that correct?

What the third party is actually doing is taking an estimate for the total population and multiplying by the relative size of the subset. They are also issuing confidence intervals for these estimates (I am not sure exactly how, my assumption is that they taking these from the total population). My second question is whether this is correct? Or does it depend on the dataset? If so, how? I understand this question is harder to know without details but my concern is really understanding whether this is a good idea or not in theory.

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    $\begingroup$ This is finite population sampling if you take the original data set as the population. In that case a random sample would allow inference about the population parameters with the uncertainty due to the sample of size n. I agree that it is problematic to try to draw inference about specific subsets of this population. $\endgroup$ – Michael Chernick Apr 14 '17 at 22:01
  • $\begingroup$ For some reason, I am unable to answer under my original name and I can't add any comments... Thanks for wading through my question. I am unsure exactly what you mean by units of the subset never being selected so I will try and give some more detail to clarify. Yes, it is theoretically possible that there are no "units" from the subset of population in the sample. I strongly suspect that this isn't the case but..I think what you are saying is that it doesn't matter? That the only way to make inferences is to draw a sample only from the population you are interested in? My interest is very muc $\endgroup$ – ConfusedAboutStats Apr 15 '17 at 16:21
  • $\begingroup$ Below you give the example "To expand slightly, imagine ... we wanted to know about GDP of a certain region". You are talking about a biased sample here, so you cannot expect to extrapolate between subsample and population (e.g. consider the map here). $\endgroup$ – GeoMatt22 Apr 15 '17 at 17:13
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Under a finite population sampling framework and if the sample is selected randomly according to a certain scheme, it is possible to infer population parameters from sample observations.

However, if my understanding of your question is correct, the units of that particular subset of the population are never going to be selected. That is, some units have zero probability of being included in the sample, which violates an important condition a sampling design must satisfy.

Thus, under these circumstances, inference would not be reliable, as it would be carried out over a population which is not the one the sample has been selected from.

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