Prediction intervals and confidence intervals are both used with regression models, but they answer different questions. A confidence interval estimates where the average response is likely to be for a given x value. A prediction interval estimates where one new individual data point is likely to fall for that same x value.
This distinction matters because predicting an individual outcome is more uncertain than estimating an average.
Key Facts
- A confidence interval for the mean response estimates E(y | x0), the average y value at a chosen x0.
- A prediction interval estimates ynew, one future individual response at a chosen x0.
- In simple linear regression, the fitted value is yhat = b0 + b1x.
- A typical confidence interval has the form yhat0 ± t*SEmean.
- A typical prediction interval has the form yhat0 ± t*SEpred, where SEpred is larger than SEmean.
- Prediction intervals are wider because they include both uncertainty in the estimated mean and random scatter of individual data points.
Vocabulary
- Confidence interval
- A range of plausible values for a population parameter, such as the mean response at a given x value.
- Prediction interval
- A range of plausible values for one future individual observation at a given x value.
- Mean response
- The average value of the response variable y for all individuals with the same x value.
- Residual
- The difference between an observed value and the value predicted by the regression model, written as residual = y - yhat.
- Standard error
- A measure of the typical uncertainty in an estimate or prediction.
Common Mistakes to Avoid
- Using a confidence interval to predict a single future value is wrong because a confidence interval only describes uncertainty in the mean response, not individual scatter.
- Thinking a 95% interval means there is a 95% chance the specific computed interval contains the value is wrong in frequentist statistics because the interval is random before sampling, while the true parameter or future value is fixed after the data are collected.
- Assuming the two intervals have the same width is wrong because prediction intervals include extra variation from individual outcomes around the regression line.
- Ignoring the x value where the interval is made is wrong because intervals usually get wider farther from the center of the observed x values.
Practice Questions
- 1 A regression model predicts yhat = 12 + 3x. At x = 4, the standard error for the mean response is 2 and the critical value is t* = 2.1. Find the 95% confidence interval for the mean response.
- 2 For the same model, at x = 4 the standard error for predicting an individual future value is 5 and t* = 2.1. Find the 95% prediction interval for an individual response.
- 3 A scientist wants to estimate the average test score of all students who study 6 hours, while a teacher wants to predict one new student's score after studying 6 hours. Which interval should each person use, and why?