I am trying to solve a probability/statistics problem at work, and it's been a while since I got my hands dirty. So your help will be much appreciated.
Given $n$ normal distributed variables $X_i \sim N(\mu_i, \sigma_i)$, compute the distribution of the $Z$, where:
\begin{equation} Z = 1/n \sum_{i=1}^n Y_i \end{equation}
and $ Y_i = \begin{cases} 1 & a\leq X_i \leq b, \\ 0 & \text{otherwise}. \end{cases} $
with constants $a$ and $b$.
If the $X_i$ are independent, then so are the $Y_i$, and $nZ$ is distributed as the sum of $n$ independent Bernouilli rvs, with expectations $p_i =P(a\le X_i\le b)=\Phi((b-\mu_i)/\sigma_i) -\Phi((a-\mu_i)/\sigma_i) $. The one number that tells the most is the sum $\lambda=\sum_{i=1}^n p_i$. In your applications, is it tiny, huge, or what?
There is no simple general formula for the distribution of $Z$, but depending on the values of the $p_i$ and $n$, various approximations might be relevant. Very often, the distribution of $nZ$ can be approximated by a Poisson distribution with expectation $\lambda$; Le Cam's theorem bounds the approximation error. If $n$ is very large and the $p_i$ values are bounded away from $0$ and $1$, a Gaussian distribution might give a better approximation. (These cases overlap if $\lambda$ is big, so you should work out both.)
For small values of $n$ and given values of the $p_i$ you can work out the theoretical distribution of $nZ$ by mimicking the Pascal Triangle construction. This computes the convolution of the laws of $Y_1$, of $Y_2$, etc, using an update rule that makes row $i$ from row $i-1$ by adding $1-p_i$ times row $i-1$ plus $p_i$ times the shift of row $i-1$. When you are done the $n$-th row is $(P(nZ=0), P(nZ=1),\ldots)$.
Without further detail about the ranges of values of $n$ and the $p_i$, and the degree of approximation you are willing to tolerate, it is hard to give more particular advice. In one sense the tiny $\lambda$ and huge $\lambda$ cases are more mathematically tractible, and the $\lambda\approx 1$ case harder.
There is a wonderful book on Poisson Approximation covering (among other things) cases where the $Y_i$ are dependent. Depending on details of your problem, some of its results might be applicable.