I am interested in computing confidence intervals for the mean of a random variable $X$ given $N$ i.i.d. samples of $X$, where $N$ is $Binomial(n, p)$. Any time I read about confidence intervals for the mean it is assumed that the number of samples is fixed, which makes the asymptotic distribution of the sample mean Gaussian, and therefore allows for student-based confidence intervals and the like to be justified. However, if the number of samples $N$ is a random variable itself, then the ratio $$ \frac{\sum_i X_i}{N} $$ will not be necessarily normal (see, for instance, http://en.wikipedia.org/wiki/Ratio_distribution#Gaussian_ratio_distribution).

What is the best way to deal with this scenario? Will bootstrapping be theoretically justified in this case?