Confidence Interval and Prediction Interval的区别

We use a confidence interval to quantify the uncertainty surrounding the average sales over a large number of cities. For example, given that100,000 is spent on TV advertising and $20,000 is spent on radio advertising in each city, the 95% confidence interval is [10,985, 11,528]. We interpret this to mean that 95% of intervals of this form will contain the true value of f(X). On the other hand, a prediction interval can be used to quantify the uncertainty surrounding sales for a particular city. Given that $100,000 is spent on TV advertising and $20,000 is spent on radio advertising in that city the 95% prediction interval is [7,930, 14,580]. We interpret this to mean that 95% of intervals of this form will contain the true value of Y for this city. Note that both intervals are centered at 11,256, but that the prediction interval is substantially wider than the confidence interval, reflecting the increased uncertainty about sales for a given city in comparison to the average sales over many locations.

Confidence Interval是对所有observation依据模型求出的均值，而Prediction Interval是对某一特定observation的预测值。通常后者的区间大于前者，因为后者包含有irreducible error.