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The Least-Squares Regression Line (Lesson 3.6)

Chapter 3 - Day 7

Learning Targets
  • Calculate the equation of the least-squares regression line using technology.
  • Describe how outliers affect the least-squares regression line.

  • Explain the concept of regression to the mean.

Activity: What is the Relationship Between GPA and ACT? 

Experience First

Before you begin the activity, it's worth discussing with your students, "How does the computer or calculator find the line of “best” fit? What makes it the line of best fit?" Using this Desmos eTool we let students try to move the line into the “best” spot (be sure you hide the line of best fit to start). They discovered that the line of “best” fit is the one that minimizes the sum of the squared residuals (the least squares regression line!). We added these squares to the picture on the top of the activity.

For the activity, students will use the 2 Quantitative Variable applet. The first three questions are a review from yesterday’s lesson, while the last three questions get students thinking about what happens to the least-squares regression line when an outlier is introduced. 

Formalize Later

When presenting the QuickNotes for the 2nd learning target, make a clear distinction between the following two types of outliers:

  1. Outliers with relatively low or relatively high x values. These outliers “tilt” the line towards themselves (they are greedy). Think of this as a rotation of the line (like a teeter totter).  

  2. Outliers with x values near the mean of x. These outliers will “pull” the line towards themselves (also greedy), but without changing the slope too much. Think of this as being a translation of the line.

Notice that the 3rd learning target about regression to the mean is not contained within the activity or the Check Your Understanding. We gave students exercise #15 as part of their homework and will use this question to have a short discussion during the next class period. 

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