In a regression with a 95% prediction interval, which must be true?

The reasoning behind the statement that the prediction interval is always wider than the confidence interval pertains to the distinct purposes of these two intervals in regression analysis.

A confidence interval estimates the range of values within which we expect the mean of the response variable to fall for a given set of predictor variables. It reflects the uncertainty associated with estimating the mean response based on sample data. Conversely, a prediction interval expects to capture the range of possible values for a single new observation from that response variable.

Since the prediction interval accounts for both the uncertainty in estimating the mean and the variability of individual predictions around this mean, it inherently includes additional variability not present in the confidence interval. This additional width results from the fact that we are predicting a future value, which introduces more uncertainty than simply estimating the mean response.

Consequently, the nature of these intervals ensures that the prediction interval will always be wider than the confidence interval, particularly at the same confidence level, such as 95%. The other statements refer to relationships and behaviors of these intervals that may not universally apply or can be misleading without proper context.