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 teh 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

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