Interpret the standard deviation of the residuals and r-sq and use these values to assess how well a least-squares regression line models the relationship between two variables.
Describe how the least-squares regression line, standard deviation of the residuals, and r-sq are influenced by unusual points.
Activity: Can you guess my IQ?
For 12 years we have been teaching students the fill-in-the-blank exercise for interpreting r-squared:
______% of the variation in (y-variable context) can be accounted for the least-squares regression line using x = (x-variable context).
And finally today for the first time, students did an activity to discover (and hopefully better understand) this interpretation. The activity went great and time will tell if this approach will yield better results. The big idea to understand in the context of the activity is that a guidance counselor who doesn't know students' GPAs will not make as good of predictions for IQ scores than a guidance counselor who does know students' GPAs. Thus, GPA helps to explain the variation in IQ scores.
Teaching Tip: The Standard Deviation of the Residuals
Students will not be expected to calculate s by hand on the AP® Exam. Most often it will be provided in computer output. For this lesson, we had them use an applet to find s. To have the graphing calculator compute the standard deviation of the residuals, run the LinRegTTest by pressing STAT, choosing the TESTS menu, and scrolling to LinRegTTest. We recommend that you not show this to your students until Chapter 12.
When introducing the idea of the standard deviation of the residuals, remind students about the standard deviation we learned earlier. In Chapter 1, the standard deviation was the typical distance of the actual values from the mean. In this chapter, the standard deviation is the typical distance of the actual y-values from the predicted y-values.