# Can I throw a fixed bias assumption into my confidence interval?

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.