For several weeks now, I'm stuck with this problem:

Given some $x_i$ for $i\in1...n$ I have to make a prediction (by some regression algorithm such as OLS regression or regression trees, since the response is continuous) for $y_i$. Not only the 'predicted value' is necessary to know but also the lower and upper bounds in order to say that $y_i$ lies within $x$% certainty within these bounds (hence, we have to create a prediction interval I think?).

Now the difficulty comes in play:
We now have a prediction of $y_i$ including its upper and lower bounds but we eventually have to use these values to make a prediction(interval) of the total value $y=\sum_i^n a_iy_i$ (where $a_i$ is known for every $i$). I would simply think that we can just add the individual lower/upper bounds for the $y_i$ and arrive at the interval in which we can say that $y$ lies within $x$% certainty between these values?

In short: How do we create a $x$% prediction interval for $y$ by using information about the predictions of $y_i$?

Thanks in advance!