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I know such a problem is explained many times, but I have still a problem with the concept and interpretation:

I would like to estimate export weight for 2016

enter image description here

  • The red point is estimated point
  • red lines are prediction interval
  • blue lines are confidence interval

As I understand the actual export weight for 2016 is between the red lines with probability 0.95 (95% prediction interval)

and the parameter of fitted model: (here $\beta_0$ and $\beta_1$)

$$\mathit{Y}=\beta_0+\beta_1X_1+\varepsilon$$

are between both blue lines confidence interval. That means possible green lines are between both blue lines (confidence interval)

My Question:

If all possible green lines are between confidence interval, then there is not possible to have the estimated point outside the confidence interval. But we have determined, it is possible to have an estimated point between red line or prediction interval. How can I interpret it correctly

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  • $\begingroup$ Your interpretation "all possible green lines are between confidence interval," is not correct $\endgroup$
    – Glen_b
    Jul 26, 2016 at 11:14
  • $\begingroup$ To make it more confusing, the prediction interval is only 95% correct when the assumptions are 100% correct. To me, it looks like the assumptions are not correct, or at least not very useful. So the 95% prediction interval is (in practice) not really a 95% prediction interval. $\endgroup$ Jun 22, 2020 at 8:56

1 Answer 1

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The blue lines don't matter for your prediction. You can see what happen with your data, because it's the same you can expect to happen with your predictions. Please notice that some past observations fall outside of the blue lines, but most of all (about 95%) fall between red lines. That is what we should expect to happen with your predictions: although you have a single red point, the real (future observed) value is expected to fall anywhere between red lines 95% of times.

If we return to your model we can see better what both colours of lines mean. Your model is:

$$\mathit{Y}=\beta_0+\beta_1X_1+\varepsilon$$

Where $\beta_0$ and $\beta_1$ are unknown parameters we only can estimate and $\varepsilon$ is a random variable.

The blue lines are confidence intervals for $\beta_0+\beta_1X_1$, that is, intervals for the green line, but please notice that that is not the future observation (that is just your point estimation of it). Your future observation will include an $\varepsilon$ term which will cause more variability, and the red lines account for that extra variability.

Edit:

It was asked (and answered) in comments what are the blue lines useful for. For completeness, this is the answer:

If you are predicting future observations, confidence intervals (blue lines) don't have a direct use. However they can be useful sometimes. I give a couple of examples:

  • When epsilon stands for an error of measurement, we are usually more interested in predicting means than in predicting observations. Then, blue lines are more useful than red ones.
  • Blue lines show how much can we improve predictions by increasing sample size. If blue lines are nearly as apart as red lines, we could improve a lot our predictions with a larger sample; in the opposite case there is very little to gain.
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  • $\begingroup$ Thanks Pere, then I can say, that the Confidence intervals doesn't play any roll in the quality of prediction point. I cant steel understand, how does confidence interval help us? and why we need confidence interval in prediction $\endgroup$
    – Kaja
    Jul 26, 2016 at 11:38
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    $\begingroup$ If you are predicting future observations, confidence intervals (blue lines) don't have a direct use. However they can be useful sometimes. I give a couple of examples: 1) When epsilon stands for an error of measurement, we are usually more interested in predicting means than in predicting observations. Then, blue lines are more useful than red ones. 2) Blue lines show how much can we improve predictions by increasing sample size. If blue lines are nearly as apart as red lines, we could improve a lot our predictions with a larger sample; in the opposite case there is very little to gain. $\endgroup$
    – Pere
    Jul 26, 2016 at 13:35
  • $\begingroup$ Why does the red line also not expand out like the blue line? I would imagine that it should also. $\endgroup$
    – confused
    Jun 22, 2020 at 6:25
  • $\begingroup$ @confused - That might be another good question. $\endgroup$
    – Pere
    Jun 22, 2020 at 8:31
  • $\begingroup$ @confused, it does, but the range of x is too small for it to be immediately visible. $\endgroup$
    – waterlemon
    Jan 4, 2021 at 8:34

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