# Interpretation of (simultaneous) confidence band against fitted values in multiple regression

In a homework question, I am asked to interpret a figure of the confidence band and simultaneous confidence band of 95% confidence level plotted against predicted values. The confidence bands are produced from linear regression. I did some searching, but the explanation by Wikipedia only covers the situation where we have only one predictor, which is hard to generalize to multiple regression.

Here is my own attempt: since the fitted value by a linear model is simply a linear combination of predictors and 1, confidence bands against fitted values can be thought of as confidence bands against different levels of predictors. On the one hand, at each level of predictors, the probability of a point falling in the point-wise confidence band is 95%; on the other hand, the probability for a line representing the real relationship between predictors and the response to fall into the simultaneous confidence band across all level of predictors is 95%.

However, I'm not satisfactory about the answer, because "levels of predictors" is too vague and sounds like word-playing. In addition, the coefficients used to calculate the predicted values are random variables and thus prone to statistical error, which means the whole X-axis is somewhat random.

How do I interpret it?