This note discusses the theoretical foundations for the extension of the Wilson two-sided coverage interval to an estimated proportion computed from complex survey data. The interval is shown to be asymptotically equivalent to an interval derived from a logistic transformation. A mildly better version is discussed, but users may prefer constructing a one-sided interval already in the literature.
A note on Wilson coverage intervals for proportions estimated from complex samples
Kott, P. S. (2017). A note on Wilson coverage intervals for proportions estimated from complex samples. Survey Methodology, 43(2), 235-240. https://www.researchgate.net/publication/322751834_A_note_on_wilson_coverage_intervals_for_proportions_estimated_from_complex_samples