How to sample for compliance when a portion of the population is already known to be non-compliant? Say I wanted a representative sample to estimate the portion of a population that is compliant on "doing their taxes correctly". Normally, I'd do a random sample of that population to make inferences within a certain confidence level and margin of error (e.g. the proportion of those who did not do their taxes right).
But what if I already knew which people in the population didn't file their taxes (meaning, I already know they're non-compliant since they didn't submit anything at all)? Since it doesn't make sense to spend resources sampling a subgroup I already know is completely non-compliant, would I just sample then on the subgroup who did file their taxes and then investigate their tax returns to see if they were filled out correctly (i.e. they actually paid the correct amount in taxes)? How then would the results of that subsample on filers be combined with the complete subpopulation of known non-filers?
For example, if the population is 1000 people, and I know 800 filed their taxes and 200 didn't, knowing those 200 are already non-compliant, if I then sampled 385 of the 800 filers and found that 85 of them did their taxes incorrectly (85/385 = 22.1% non-compliance rate among filers), how would I then combine that information with the already known 200 non-compliant 'non-filers' to estimate the proportion of the population that is non-compliant?
 A: You have a population $P$ of size $\#P$ and a subset $M$ of non-filers of size $\#M$. Within the subset $F = P\setminus M$ of filers, which is of size $\#F = \#P-\#M$, there is a set $D\subset F$ of people that have filed their taxes incorrectly, and you have figured out their proportion $r$ within $F$, i.e. you know:
$$
r = \frac{\#D}{\#F}.
$$
You are looking for the total fraction $t$ of non-compliant individuals, i.e.:
$$
t=\frac{\#M + \#D}{\#P}.
$$
Combining those two equations, you get:
$$
\begin{align}
t &= \frac{\#M + \#D}{\#P}\\
  &= \frac{\#M + r\#F}{\#P}\\
  &= 1\frac{\#M}{\#P} + r\frac{\#F}{\#P},
\end{align}
$$
which can be read as follows: In the set $M$ of non-filers, the ratio of non-compliant individuals is equal to one, within the set of filers it is $r$, and the total ratio is the weighted mean of one and $r$ where the weights are the proportional sizes of the respective subsets.
In your example:
$$
\begin{align}
t &= 1 \frac{\#M}{\#P} + r\frac{\#F}{\#P}\\
  &= 1 \frac{200}{1000} + 0.221 \frac{800}{1000}\\
  &= 1 \cdot 0.2 + 0.221\cdot 0.8\\
  &\approx 0.377.
\end{align}
$$
