I am interested in computing confidence intervals for the mean of a random variable $X$ given $\require{cancel}\xcancel{N \text{ i.i.d. samples}}$ an i.i.d. sample of $N$ copies of $X$, where $N$ is $\operatorname{Binomial}(n, p)$. Any time I read about confidence intervals for the mean it is assumed that the ~~number of samples~~size of the sample 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~~ size $N$ of the sample 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?