I have a large database of binary decisions (accept or reject), broken down by state of the applicant, so that for each state I can calculate a proportion of positive decisions. e.g.
New York - 0.21% accepted Washington - 0.17% accepted California - 1.25% accepted Oregon - 2.01% accepted Alaska - 0.54% accepted etc...
As you can see, the proportion accepted is sometimes very low, and therefore if I calculate confidence intervals around them (using a binary distribution to bound the intervals between 0 and 1), the interval width can be quite large.
I want a way of determining which states have a particularly high proportion or accepted decisions. As a start, I calculated the overall national proportion, with confidence intervals, and looked whether the confidence intervals of each state overlapped with the overall national confidence intervals. If they didn't I thought this would indicate that state had a proportion significantly higher or lower proportion than average.
However, I'm worried that the overall national proportion might be disproportionately influenced by certain states where a large number of decisions were made.
Also, is my approach a bit simplistic? Should I instead use logistic regression to see whether state is a significant predictor of decision, and if so, do a post-hoc test to see which states differ? If so, what post-hoc test could I do?