I have a couple of questions for prediction and tolerance intervals.

Let's agree on the definition of the tolerance intervals first: We are given a confidence level, say 90%, the percentage of the population to capture, say 99%, and a sample size, say 20. The probability distribution is known, say normal for convenience. Now, given the above three numbers (90%, 99% and 20) and the fact that the underlying distribution is normal, we can compute the tolerance number $k$. Given a sample $(x_1,x_2,\ldots,x_{20})$ with mean $\bar{x}$ and standard deviation $s$, the tolerance interval is $\bar{x}\pm ks$. If this tolerance interval captures 99% of the population, then the sample $(x_1,x_2,\ldots,x_{20})$ is called a success and the requirement is that 90% of the samples are successes.

Comment: 90% is the a priori probability for a sample to be a success. 99% is the conditional probability that a future observation will be in the tolerance interval, given that the sample is a success.

My questions: Can we see prediction intervals as tolerance intervals? Looking on the web I got conflicting answers on this, not to mention that nobody really defined the prediction intervals carefully. So, if you have a precise definition of the prediction interval (or a reference), I would appreciate it.

What I understood is that a 99% prediction interval for instance, does not capture 99% of all future values for all samples. This would be the same as a tolerance interval that captures 99% of the population with 100% probability.

In the definitions I found for a 90% prediction interval, 90% is the a priori probability given a sample, say $(x_1,x_2,\ldots,x_{20})$ (size is fixed) and a single future observation $y$, that $y$ will be in the prediction interval. So, it seems that both the sample and the future value are both given at the same time, in contrast to the tolerance interval, where the sample is given and with a certain probability it is a success, and under the condition that the sample is a success, a future value is given and with a certain probability falls into the tolerance interval. I am not sure if the above definition of the prediction interval is right or not, but it seems counterintuitive (at least).

Any help?