Determine which of the following statements is/are true for a simple linear relationship,

[[math]]y = \beta_0 + \beta_1x + \epsilon[[/math]]

• If [math]\epsilon = 0[/math], the 95% confidence interval is equal to the 95% prediction interval.
• The prediction interval is always at least as wide as the confidence interval.
• The prediction interval quantifies the possible range for [math]\operatorname{E}(y | x).[/math]

• I only
• II only
• III only
• I, II, and III
• The correct answer is not given by (A), (B), (C), or (D).

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key E

I is true. The prediction interval includes the irreducible error, but in this case it is zero.

II is true. Because it includes the irreducible error, the prediction interval is at least as wide as the confidence interval.

III. is false. It is the confidence interval that quantifies this range.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.