Say I'm doing a confidence interval for a proportion $p$. What if I make the assumption that the sample is biased in a systematic way, by at most 10% say (ad-hoc/arbitrary choice) -- i.e. that the sample comes from a population whose proportion is in the range $(p/1.1, 1.1p)$.
Then I think the confidence interval would just be $(l/1.1, 1.1r)$ in place of $(l, r)$.
This feels quite silly. It may even be incoherent (let me know if so).
But yeah, in practice, I might want to use a confidence interval to express uncertainty about some proportion -- and I know that my sample might be slightly biased, but I don't think it's super biased. And I want to make sure this bias uncertainty doesn't get lost when I report an interval.
Is there some better way to go about this?
Really I think my main purpose is to
(1) express an uncertain quantitative estimate of a proportion (so I naturally think conf intervals)
(2) guard against people interpreting narrow confidence intervals as more than they are
Is this a good approach for that?
When comparing multiple groups, reporting effect size maybe accomplishes this kind of thing.