Linear Regression Models

A line that predicts

When a scatterplot shows a linear form, you can summarize the relationship with a linear regression model: an equation that uses the explanatory variable $x$ to predict the response variable $y$.

Here $a$ is the y-intercept, $b$ is the slope, and $x$ is a value of the explanatory variable. What $a$ and $b$ mean in context is the subject of 5.5 Least-Squares Regression.

The hat on $\hat{y}$ (say "y-hat") marks a predicted value. Plain $y$ is a value that was actually observed. Keep the two apart. The model gives a prediction, and real cars land above it or below it.

The model only makes sense when the form looks linear. A straight line fit to curved data, like the typing speeds in 5.1 Graphical Representations Between Two Quantitative Variables, will miss in the same direction over whole stretches of the data.

Making a prediction

To predict, substitute a value of $x$ and calculate. For a 6-year-old sedan:

The model predicts an asking price of about 17,030 dollars. Two of the sedans in the data are 6 years old, listed at 16,800 and 17,800 dollars. Neither matches the prediction, and the model never promised that it would. 5.4 Residuals measures how far each real car lands from the line.

Inside the data or outside it

The 14 sedans are 1 to 10 years old. A prediction for an age inside that interval, like 6 years, is interpolation. A prediction for an age outside it, like 20 years, is extrapolation.

Interpolation has data on both sides of it. Extrapolation assumes the straight-line pattern keeps going where nobody collected any data, and patterns often change. Cars lose value quickly at first and more slowly later, so a line fit to younger cars drops below zero long before real prices do. The model hits zero at about 17 years.

The further you extrapolate, the less reliable the prediction. A 12-year-old sedan is a small stretch past the data. A 20-year-old sedan is a leap.