Linear Regression and Residuals

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SWBAT create residual plots and use them to determine if a linear model is an appropriate for a given two-variable data set.

Big Idea

Examining the size and distribution of errors made by a model can help us determine if the model is appropriate.

Warm-up: How Close was the Prediction?

20 minutes

To introduce the idea of residuals, I ask my students to calculate the prediction errors of two models. In Warm up Prediction Errors, students use models to calculate a predicted value of output and then compare the predicted output to an observed value [MP4].

When students have completed this warm-up, I ask if they think the other values predicted by the model would be too high too low or just right.  In this way, I start a conversation about the relationship between the distribution of residuals and the appropriateness of a given model. Specifically I hope that they will agree that a good model will sometimes over-predict and sometimes under-predict, and that the errors should be about the same size for small and large values of output.  


Direct Instruction

20 minutes

I begin this note-taking session by reminding my students of the three conditions for using the linear model.

  1. the data set includes paired values of two quantitative variables
  2. the scatter plot displays a linear form
  3. there is no pattern in the residual plot when the linear model is applied

In the previous lesson, we discussed the first two conditions and this lesson will cover the third.  In the warm-up, my students practiced calculating prediction errors using models that they encountered in previous units of study (exponential and polynomial).  I use this work as a springboard for discussing how residuals are used to determine if the linear model is a good fit.  

The final test of whether it is appropriate to use a linear model is to create a plot with the residuals on the y axis and the input values on the x-axis and examine the plot for patterns.  If it appears that there is regularity to the residual plot, we can conclude that the linear model is NOT a good fit.  That is, we want to see very scattered residuals before accepting the linear model as appropriate (MP4).

Practice Calculating Residuals

40 minutes

Students practice calculating and interpreting residuals by completing WS Calculating Residuals with their table partners.  This activity sheet provides students with two data sets to work with.  Their job is to use the graphing calculator to find the line of best fit and calculate residuals.  Although students may do each of the calculations separately, I guide them towards the use of a spreadsheet page on their calculator for these repeated calculations (MP5).

Close and Assignment

10 minutes

When students have completed the worksheet, we come together as a class to discuss the results.  I ask volunteers to share answers to each part, with students coming to the board as necessary.  I allow time for questions and then pass out HW Calculating Residuals, which is the homework assignment for the evening.