Prediction interval I calculated an 80% prediction interval of the outcome of interest (proportion of patients, average temperature etc.) based on my previous study. Is it true that in the future study I will receive the outcome within the calculated prediction interval with 80% probability? Which means that if I do a billion studies the outcome will be within the prediction interval in about 80% of them?
 A: The answer is no: no such (nearly) free lunch can exist, even if the
model is perfect.
Consider for instance a linear regression using $n$ observations
$[x_i,\,Y_i]$. Given a "new" design point $x^\star$ you can compute a
$80\%$ prediction interval; It contains one new $Y^\star$
corresponding to $x^\star$ with probability of $80\%$. But it does not
contain $80\%$ of a large number of new values $Y^\star_j$ made at
$x^\star$, even if the model is perfectly adequate. As suggested by
the formulation of your question, you can not guess exactly the
distribution of an infinite number of new values from a finite number
of observations.
To reach the wanted probability, you have to draw a new dataset with
observations $[x_i, \, Y_i]$ for each prediction that you make. This is
well explained by G.W. Senedecor and W.G. Cochran in the regression
chapter of their famous Statistical Methods book.
An alternative where the expected coverage rate holds is when the
prediction is updated sequentially, thus modifying the prediction
interval: the first new couple $[x^\star_j, \,Y^\star_j]$ is included in
the data, then the model estimates and the prediction interval are updated
before a new prediction is made. Again, this must be repeated for each
prediction, and the coverage will be reached only in the long
run. This context is classical in time-series analysis.
A: It is probably not true. In an ideal world, where observations are truely i.i.d. and your first study had a truely representative sample and all the other assupmtions (distributions etc.) were true, than yes. However, in real life all those assumptions fail to be true and thus you will probably face larger variances then expected. 
All of this is of different importance in different fields. It may be possible, to do the same experiement twice in physics - but then again, probably in fields that use statistics only rarely. In fields like medicine, psychology, sociology, the idea of a second sample that is identically distributed as the first sample, most of the time remains a dream.
A: Thanks for your answers. According to the citation below (book: Statistical Intervals A Guide for Practitioners and Researchers) the answer is Yes. Suppose that, based on the x = 20 nonconforming integrated circuits from n = 1,000 randomly selected units, the manufacturer desires a 95% prediction interval to contain the number of nonconforming units in a future sample of m = 1,000 randomly sampled units from the same production process. The conservative method given by (6.9) gives [Y,Y] = [9, 35] for a conservative 95% prediction interval and Y = 32 for an upper conservative 95% prediction bound for Y . Thus, based on x = 20 nonconforming units from the n = 1,000 sample units, one can, for example, assert, with (at least) 95% confidence, that the number of nonconforming units in the future sample of m = 1,000 will not exceed 32 units.
