Imagine a race with 4 relay runners running a 4x100m. I have different bits of information on how fast each runner can complete various distances which I use to predict how quickly they can run 100m. Each runner has a regression model which predicts their expected finish time and an interval in which I am 95% confident they will finish (the predcition interval). How would I calculate the prediction interval for the finish time of the whole race (ie all 4 runners together)? I have the Standard error of the regression for each runner's model and the standard error of the prediction (so I can calculate each individual prediction interval).

Any help greatly appreciated!