I am conducting a linear regression: $Y=\alpha+\beta\times X+\epsilon$, $\epsilon\sim N(0,\sigma^2)$. It turned out that the confidence interval for the predicted mean $Y$ was really small (figure 1), that it was hardly differntiable from the predicted mean.
On the other hand, we can see that the observed $Y$s actually spread quite widely around each X. The second figure shows the predicted interval, which is large.
My question is:
- What does this model tell me? Does it mean that it can be a perfect model to predict the mean? since the confidence interval in figure 1 is small.
- On the other hand, figure 2 tells that the random error in this model is high. Is there anything we can do (or is it necessary) to reduce such random error, although we already have a "perfect" model? For instance, will adding extra useful (assume) variables help further explaining (reducing) the random error?
Instead of asking what's the difference between these two types of intervals (see @whuber's link), I am interested in if a small confidence interval and a large prediction interval exist, what can we say and what can we do about such a model? is such model the best already and we should submit the result? Or something can still be done to further explain the random error? Can someone help me explaining this result?