Practice a College Board-style free response question on interpreting a least-squares regression model. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 5 · Least-Squares Regression
A used-car dealer studies the relationship between a car's age (in years) and its price (in dollars) for a sample of 15 cars of the same model. A least-squares regression is fit, giving the equation predicted price = 21,500 − 1,450(age). For this model, r² = 0.83, and the ages in the sample range from 1 to 12 years.
A
Interpret the slope of the regression line in the context of this problem.
✓ Model answer
The slope is −1,450. For each additional year of a car's age, the predicted price decreases by $1,450, on average.
Why it scores: States the slope value AND interprets it as a predicted (average) change in price per one-year increase in age, in context with units. Saying price "decreases by $1,450" without "predicted/on average" or without the per-year rate would lose credit.
B
A particular 6-year-old car in the sample actually sold for $11,500. Find the predicted price for a 6-year-old car, then calculate and interpret the residual for this car.
✓ Model answer
Predicted price = 21,500 − 1,450(6) = 21,500 − 8,700 = $12,800. The residual is observed − predicted = 11,500 − 12,800 = −$1,300. Because the residual is negative, the model overpredicted this car's price by $1,300 — the car sold for $1,300 less than the line predicts.
Why it scores: Computes the predicted value, applies residual = observed − predicted (correct order), AND interprets the negative sign as an overprediction in context. Reversing the subtraction is the most common error.
C
Interpret the value of r² in context, and explain why using this model to predict the price of a 25-year-old car may be unreliable.
✓ Model answer
About 83% of the variation in the price of these cars is explained by the linear relationship with their age. Predicting the price of a 25-year-old car is extrapolation: 25 years is well outside the observed age range of 1 to 12 years, so the linear pattern may not continue that far and the prediction is unreliable (it could even give a nonsensical negative price).
Why it scores: Interprets r² as the proportion of variation in price explained by the linear relationship with age (in context), AND identifies predicting at age 25 as extrapolation beyond the data range, explaining why that is a problem.
How to score points on AP Statistics FRQs
Interpret slope and intercept in context. Use "predicted" and "on average," and include the variables and units.
Residual = observed − predicted. Keep the order straight, and say whether the model over- or under-predicted.
Interpret r² as explained variation in the response variable, not as points on the line or as the correlation.
Flag extrapolation. Predictions outside the data's x-range are unreliable — say so.
Check the linear model with a residual plot, and never claim causation from a regression on observational data.