# Estimating a tail probability based on the sample mean and standard deviation

I want to know if a sub-population has the same upper quantile as a parent population. Assume that in the parent population, proportion $1-\alpha$ of the data lies below threshold $k$. I can observe the population mean and variance, and the tail probability $\alpha$ of the threshold. Let $z_{\alpha}$ be the associated normal quantile.

Suppose I have a sample from the sub-population of size $n$, and I observe $\bar{X}$ and $S$. If I assume that the random variable in the population and the sub-population is normally distributed, then a sensible criterion would be

$$\bar{X} + z_{\alpha}S \leq k$$

If the expression exceeds the threshold, then I conclude that the sub-population is not behaving according to specification. However, I don't like this criterion, since it fails to account for sampling variation in $\bar{X}$ and $S$. An improvement might be to account for variation in the sample average and use:

$$\bar{X} + z_{0.95}\frac{S}{\sqrt{n}} + z_{\alpha}S \leq k$$

where $z_{0.95}$ is the 95th percentile of the normal, or some such confidence value.

Since data in the sub-population could be less variable than in the overall population, I don't want to make assumptions about the variances. I just need for proportion $1-\alpha$ of the sub-population to be below the threshold. It's not enough to establish that the sub-population mean is significantly different from the population mean. I want an inference about the upper quantile.

Am I on the right track here, and can someone suggest a better way to establish a threshold on the sub-population?