# How to analyze a prediction interval inside another prediction interval

For those who know something about statistics,

I'm wondering how to analyze a prediction interval encompassing other prediction intervals. To provide some background, I'm not talking about confidence intervals (they are often confused). I'm computing a 95% one-sided prediction upper-limit for a set of data. The definition I'm familiar with for a prediction upper-limit is that we can predict that a future test would be less than the upper-limit 95% of the time. Without assuming anything about the data distribution, the maximum of 19 instances will give me a 95% upper limit. This is because 19 data points create 20 bins, and so a future test (#20) has a probability of 1/20 or 5% of landing within each bin. The last bin is everything higher than the maximum value of the 19 repetitions, and so there is a 5% chance the future test will be higher than this value, and a 95% chance it will be lower.

So that's all fine and dandy, but now I have two levels of prediction limits, and the best way I can illustrate this is with a simple example. Suppose I want to analyze the performance on my computer, and there is inherent randomness in how I test this, so I create a 95% prediction upper-limit using 19 tests to know a future test has a 95% chance of falling below that value for my computer. But there are many different types and brands of computers, and if I want to know about the performance of computers in general, I test 18 more computers (19 total) of different types and brands. So now each computer has its own 95% prediction upper-limit specific to itself, and the maximum of all 19 upper-limits provides an upper-limit for computers in general.

My question is this: Do I have a 95% prediction limit for all computers? Because each one has an individual 95% limit, is it actually a 90.25% limit (.95 * .95)? Or possibly a 99.7% limit (since there are 19*19 total data points)?

Thanks,

Tyler