More Americans admitting atheism

[CliveStaples]

I would expect that P(X doesn't respond) would consist of both P(X is apathetic about religion) as well as P(X is apathetic about questionnaires), amongst other reasons.

Right, which is why I gave the inequality that I did.

X is apathetic = (X is apathetic AND X responds to the questionnaire) + (X is apathetic AND X fails to respond to the questionnaire)

Since (X is apathetic AND X fails to respond to the questionnaire) is a subset of (X fails to respond to the questionnaire), P(X is apathetic AND X fails to respond to the questionnaire) <= P(X fails to respond to the questionnaire).

So under the assumption that every respondent is being truthful and honest (if they say they believe p, then they in fact believe p), and that all responses are properly tallied and so on, then I think we can conclude that:

P(X fails to respond) + P(X responds as apathetic) = P(X fails to respond AND X is apathetic to religion) + P(X fails to respond AND X isn't apathetic to religion) + P(X responds AND X is apathetic to religion) >= P(X fails to respond AND X is apathetic to religion) + P(X responds AND X is apathetic to religion) = P(X is apathetic to religion).

That is, P(X fails to respond) + P(X responds as apathetic) >= P(X is apathetic to religion).

How many random surveys have measured religious belief and included an option sufficiently similar to apathy? We can use them to get an upper bound on the number of people who are apathetic about religion (with confidence dependent on the sample size of the surveys used).