Estimating the proportion of observations inside a given interval I have a question which should be a failry common question, but for which I can not find any reference.
For simplicity, we will assume that our variables are all i.i.d and normally distributed.
I take a sample of some incoming parts, and measure a dimension. I obtain a mean m, and a standard deviation s. I also have an engineering specification which says that this dimension should be between LL and UL (lower and upper limits). I am trying to figure out, based on what I measured on the sample, the interval (min and max) proportion of parts I could expect to be "in spec". I.e. some interval which says that at least min% of the parts  can be expected in spec, and at most max% of the parts can be expected to be in spec. I could use the standard normal distribution, but this assumes that I know the sample mean and standard deviation, which I do not (only have an estimate). So what I compute with the Z distribution will underestimate [min, max], as it does not account for the expected variability in the measured m and s statistics.
I am not trying to get a tolerance interval (based on the normal distribution): that would tell be that P% will be within a given interval at a certain confidence level. I am trying to get basically the reverse: given an interval [LL, UL], what percentage of observations could lie in this interval, at a given confidence level.
It would seem to be a typical problem for any manufacturing operation, but I am drawing blanks?
Thaks

This is basically a variation of the prediction interval. Consider your sample of incoming parts as independent realisations $(x_1, \cdots, x_n)$ generated from some process $X\sim N(\mu, \sigma)$ (let's assume it's normally distributed).
From them, you calculate the sample mean $\hat\mu$ and the sample standard deviation $\hat \sigma$ according to the standard formulae. We now have an estimate of $X$. This is $\hat X \sim N(\hat\mu, \hat\sigma)$.
We are then interested in knowing the proportion of parts which lie within the interval $(\text{LL}, \text{UL})$. This is the simple calculation $Pr(\text{LL} < \hat X < \text{UL})$, which you can calculate easily from most statistical packages.
This is probably good enough to get by with. We have ignored the fact that $\hat\mu$ and $\hat\sigma$ each have uncertainty associated with them. If this worries you, you could do a bootstrap to account for this. But if your sample size $n$ is large enough it's probably not worth worrying about.
