I am trying to provide an interval estimate for the 0.8-quantile of some numeric data, which is assumed to be an IID sample from some unknown, continuous distribution. I constructed an Empirical CDF for my data (black line in the image below). I used the information contained in this Wikipedia article and this review to construct point-wise 95% confidence intervals for the true CDF at many values of x. (These intervals show up as the red lines in the image below. Note that I did NOT find simultaneous confidence bands.)
How do you turn bounds on the CDF into bounds on quantiles? I found where the bounds on the CDF reach 0.8. Can I take the x-values of these intersections as bounds for a 95% confidence interval on the 0.8-quantile? In other words, is [28 000, 36 000] a valid 95% CI for the 0.8-quantile? Is it valid to invert the bounds on the CDF in this manner in order to obtain a confidence interval for a quantile?
Using @BruceET's method, the CI for the 0.8-quantile would be [28 500, 36 500]. I think that my approach in the blue lines finds one-sided confidence intervals at different values of x and then pieces those together to find a two-sided CI for the 0.8-quantile. @BruceET's method seems more straightforward. Would one be preferred to the other? More importantly for the purposes of this post, what makes it valid to transform a CI for the CDF into a CI for the p-quantile of the population?