An interval for the mean and an interval for a new day
From a fitted linear model, one interval covers the mean ozone at a stated temperature and wind speed, and another covers a new day’s ozone at those same values.
Why is the second wider, and which one is the forecast for a single new day?
The mean response at a fixed covariate value is a parameter of the fitted model. Its interval only has to cover that mean, so the width comes from uncertainty in the coefficients. A new day has that uncertainty and also a new residual. The prediction interval adds the residual variance, and it is wider. A forecast for one new day is the prediction interval. Quoting the narrower interval as a range for one day’s ozone understates the uncertainty the model itself claims. Neither interval is a causal effect of setting the weather. Both are statements inside the fitted line.