Samples Tell a Lot about Intervals

Confidence intervals in multiple regression models can be used to describe the uncertainty surrounding how a model fits across the whole sample population. Prediction intervals are wider than confidence intervals and are used to describe a subset population from the whole sample population. One way to describe this is how the chapter described it, confidence intervals include reducible error where prediction intervals include reducible error and irreducible error; however, this means little to most everyone outside statistics.

A dataset is made of samples and can come from any population. A subset is a group within the dataset. In the chapter they used data from a company that showed the effect of advertising on sales. Such samples from a national company has a national population that can be predicted by confidence intervals. If that company wanted to predict the effect of advertising on sales from the Phoenix-metro area or any other subset of that data, prediction intervals would be used. If another company was a small-business with a few physical stores that only served the Phoenix-metro area, the samples would come from the Phoenix-metro population and be described by confidence intervals. If this small-business wants to predict the effect of advertising on sales from a specific store’s subset, prediction intervals would be used.

These two scenarios (a national company and regional small business) reveal two important details. The national company makes it clear how sample populations differ from subset populations. The small business makes it easy to imagine the nuances of predicting subsets from a sample population — people and ads easily cross borders which means making predictions of smaller groups from a dataset that describes a much larger population is more difficult. Cross advertising is one of many plausible reasons that the prediction interval is so much wider than the confidence interval.