Predictions, Residuals, and Interpreting a Regression Line
Chapter 3 - Day 4 - Lesson 3.2
Make predictions using regression lines, keeping in mind the dangers of extrapolation.
Calculate and interpret a residual.
Interpret the slope and y intercept of a regression line.
Determine the equation of a least-squares regression line using technology or computer output.
Activity: How good are the predictions for Barbie?
Students use the online applet to find the line of best fit for some Barbie Bungee data collected by one of the groups. The group forgot to record a value for 5 rubber bands, so students will use the line of best fit to make a prediction. It is then revealed that the group found their measurement for 5 rubber bands, leading students to think about how close their prediction was to the actual value (residual!).
Notice that in the activity, we avoided "formal" language (residual, extrapolation). This is done intentionally to keep the activity accessible to all students. We always layer on the formality when we debrief the activity. Remember: Experience first, formalize later.
We let students write their own interpretation for slope and y-intercept when they were working on the activity. During the debrief, we formalize by dialing in the necessary components of interpreting slope and y-intercept.
Students are most familiar with slope intercept form for equations of lines (y = mx + b). So why do statisticians prefer
y = a + bx? The answer is that most often in the real world, there are more than one explanatory variables that can help predict the response variable. Statisticians might create a model that had three explanatory variables (x1, x2, x3) that looks like this:
y = a + bx1 + b2x2 + b3x3. The y-intercept (a) is a starting point for making a prediction for the response variable and then each time we add one more explanatory variable we are refining that prediction. This process is called multiple regression (not part of the AP® Statistics course).
Always have students use context rather than x and y when writing out a regression equation. Also, make sure they don’t forget the “hat” on the response variable, or the word “predicted” in front. This will help them with calculating residuals and interpreting slope and y-intercept.
Students will often mix up the order in a calculation of a residual by taking (predicted y – actual y). An easy way to remember the correct order of subtraction is to think AP = Actual − Predicted. They should be able to remember this one.