My aim is to find the 95% confidence interval of the ratio of two variables for which I have summary statistics. More specifically, I have the prevalence of mothers drinking during their pregnancy (assumed to follow a normal distribution) and I have the prevalence of children born with foetal alcohol syndrome (FAS, supposed to follow a binomial distribution for which I know all parameters). By dividing the former by the latter I obtain the number of pregnant women that drink per 1 birth with FAS.
I thought the easiest way would be to sample both distributions N times, therefore generate N samples of this ratio and the 2.5% and 97.5% quantiles should give me the right CIs. The mean of this ratio is very far from the ratio of the means of the initial distribution though. This is a known thing and arises from Jensen's inequality (I assume this comes simply from the nonlinearity of the function f(x)=1/x). How do I interpret my Monte Carlo samples? Are they representative of the error?
The literature now proposes to use Taylor series approximations to estimate the variance of the ratio mean of 2 normally distributed variables. This would generate a variance around the ratio of the means and make it all nice and symmetrical. Given that these approaches will not give the same results at all, one giving you symmetric CIs, the other one giving you asymmetric results, which one is correct?