What is the formula (if it exists) for the sample variance / confidence interval of a quantile / percentile of the normal distribution?
For example, the 5th percentile for a standard normal population distribution is -1.64, but what is the 95% one-sided confidence interval if I have a sample of n=1,000? I.e. on average the 5th percentile of a standard normal sample will be -1.64 and 95% of the time the sample 5th percentile of a sample of n=1,000 will be below -1.54 (approximate from simulations).
There seems to be some kind of formula as it's an option in SAS (
CIQUANTNORMAL) and I think it may be related to the Probit but haven't found an explanation.