Linear Regression - Confidence interval for mean response vs prediction interval

I understand the concept of a confidence interval for the mean response (fitted line) for simple linear regression $$y$$ = $$\beta_{0}$$+$$\beta_{1}X_{i}$$. It is that taken over many times, with 95% probability the CI will contain the true regression line.

However, I am not sure how to compare this to a Working-Hotelling confidence band. The latter has the interpretation that it contains "the entire true regression line with probability 1-$$\alpha$$

Could someone explain how the two are different? thank you.

• Are you sure you mean a prediction interval? Confidence intervals can be shrunk down to be arbitrarily narrow as the sample size increases and standard errors decrease (at least under some rather routine assumptions), while prediction intervals cannot be shrunk down arbitrarily small without decreasing the residual variance toward zero. The problem seems to be more of one about confidence interval vs band than confidence vs prediction. Is that accurate?
– Dave
Aug 21, 2023 at 11:17
• The space of possible lines is two dimensional. Thus, the description in the first paragraph makes no sense and has to be re-interpreted either as a CI for the slope alone or as saying the same thing as the second paragraph. Either re-interpretation answers the question.
– whuber
Dec 22, 2023 at 19:55