Return to Question

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:

1. 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.
2. On the other hand, does it mean 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?
3. In this case, will adding extra useful (assume) variables help 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 about such a model? is such model the best already? Can someone help me explaining this result?

Thanks

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:

1. 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.
2. 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?

Thanks

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:

1. 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.
2. On the other hand, does it mean 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?
3. In this case, will adding extra useful (assume) variables help reducing the random error?

Can someone help me explaining this result?

Thanks

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:

1. 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.
2. On the other hand, does it mean 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?
3. In this case, will adding extra useful (assume) variables help 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 about such a model? is such model the best already? Can someone help me explaining this result?

Thanks