I’ve recently encountered specific problem and I am ashamed to admit that I am quite stuck. Suppose that I have random sample of data for which I want to calculate e.g. 95th quantile. Suppose that it is reasonable to fit normal distribution $N(\mu, \sigma^2)$ for this sample. I need to estimate parameters $\mu$ and $\sigma^2$ first in order to calculate the quantile (suppose that they are uknown in practice). The straightforward way is then to obtain quantile of normal distribution with estimated parameters. But I’ve encountered question whether the quantile cannot be calculated by using the Student’s distribution quantile such as $$ \overline{x} + t_{0.95}\cdot s, $$ where $\overline{x}$ is sample mean and $s$ is sample standard deviation. I am failing to justify why I can’t use the Student’s quantile (apart from argument that it is different distribution) since it resembles me the idea of confidence interval construction (where the upper bound can be perceived as quantile) for the expected value parameter $\mu$ (even though I know this is not the goal now). In this situation I also use the estimate for unknown variance parameter and, therefore, it intuitively leads me towards the Student’s distribution. Could somebody provide me with different insight? Thank you!