1

Assume the linear regression $Y=b_{0}+b_{1}X$

The $100(1-a)$% prediction interval for $x=x_{*}$ is given by

enter image description here

whereas the

The $100(1-a)$% confidence interval for $E(Y|X=x_{*})$ is given by

enter image description here

What I don't understand is the difference in the shape of the bands. The upper and lower 100(1-a)% prediction intervals are straight lines, whereas the the 100(1-a)% confidence intervals for $E(Y|X=x_{*})$ are curves, with the minimum band width at $E(X)$

I understand the reasons why prediction bands are wider than the equivalent confidence bands.

But, is there an intuitive way to understand why the shapes of the confidence and prediction bands are different?

enter image description here

The formulas are from Devore, 2017

ECII
  • 1,791
  • 2
  • 17
  • 25
  • The fundamental shapes are not different: Both are hyperbolic. It's just that in this example the prediction intervals *appear* linear but they're really not. – COOLSerdash Nov 08 '20 at 16:59
  • Thank for the link. So adding the s^2 term makes the difference in widening the bands and flattening the curvature? – ECII Nov 08 '20 at 17:19

0 Answers0