Trish runs a regression on a data set of n observations. She then calculates a 95% confidence interval [math](t, u)[/math] on [math]y[/math] for a given set of predictors. She also calculates a 95% prediction interval [math](v, w)[/math] on [math]y[/math] for the same set of predictors.

Determine which of the following must be true.

• [math]\lim_{n \rightarrow \infty} (u-t) = 0[/math]
• [math]\lim_{n \rightarrow \infty} (w-v) = 0[/math]
• [math]w-v \gt u-t[/math]

• None
• I and II only
• I and III only
• II and III only
• The correct answer is not given by (A), (B), (C), or (D).

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

Key: C

I is true. A confidence interval reflects the error in estimating the expected value. With an infinite amount of data, the error goes to zero.

II is false. A prediction interval also reflects the uncertainty in the predicted observation. That uncertainty is independent of the sample size and thus the interval cannot go to zero.

III is true. The additional uncertainty in making predictions about a future value as compared to estimating its expected value leads to a wider interval.

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