Solved – Interpreting exp(b)

Let's imagine your response variable is percent urban, and that the mean of this variable in Wyoming is $.06$. Since each person lives in an urban area or not, this means that $6%$ of the population lives in an urban area. You fit a logistic regression model to predict percent urban based on state, where Wyoming is the reference level, and you have one other state, say Montana. The analysis returns an estimated coefficient, $hatbeta_text{Montana}$, and $exp(hatbeta_text{Montana})=.08$.

That exponentiated coefficient is an odds ratio. That is, it is a constant multiple associated with the change in the odds of 'success' (i.e., a person being urban) associated with a one-unit change in the covariate. Because, you have categorical covariates, that is the ratio of the odds of being urban in Montana to the odds of being urban in Wyoming. Since the odds of being urban in Wyoming is $.06/.94=.064$, the odds of someone being urban in Montana is $.064*.8=.00512$. Now, to get the percent urban in Montana, you convert that odds to a probability $.00512/(1+.00512)=.00509%$.

So I think you have this essentially right, you just need to convert the proportion in Wyoming to an odds before you multiply it by the odds ratio.