Residuals and Residual Plots Section 3.3.3

Starter A study showed that the correlation between GPA and hours of study per week was r =.6 –Which variable would be explanatory? Why? Based on this correlation constant, would you be confident in predicting GPA if you knew how many hours per week a certain student studied? Some of the variability in GPA can be explained by the linear association, and some comes from random effects. How much is due to the association?

Objectives Calculate the value of a residual at a point on a LSRL Create a residual plot on the calculator and draw a conclusion about the goodness of fit of the LSRL

Archaeopteryx Again Look once again at the FEMUR vs HUMER lists –We already found the equation of the LSRL and graphed it HUMER = FEMUR – –Draw the graph with scatterplot on your calculator again Based on the LSRL, predict the humerus length for a femur 50 cm long –50 x – = 56.2 cm –You could also enter Y 1 (50) to get 56.2 Now predict humerus for femur = 59 –Y 1 (59) = 67.0 –But there is an actual data point where femur=59: (59, 70) The difference between the actual value minus the predicted value is called the residual –The formula is:residual = actual – predicted –Symbolically we write:

Creating a Residual Plot Paste FEMUR and HUMER into L 1 & L 2 Define L 3 with the formula L 2 – Y 1 (L 1 ) –Notice that this is just y – y-hat, so L 3 now contains all the residuals of these data –This assumes Y 1 is the LSRL equation Set up Stat Plot 3 as a scatterplot of L 1 and L 3 Turn off Plot 1 (the data) and Y 1 (the LSRL), turn on Plot 3 and tap Zoom-9 to see Plot 3 –You are seeing a scatterplot of the residuals vs the explanatory variable –This is called a residual plot

What Does it Mean? If the data were perfectly linear, the residuals would all be zero. –Then the residual plot would be points exactly on the x axis If the data miss the LSRL due to random variation, we expect the residuals to be randomly distributed. –Some will be positive, some negative –There should be no apparent pattern If the data miss the LSRL due to some curve in the data, the residuals will show a pattern. –Where the LSRL goes through the data, residuals will be small –Elsewhere the residuals will be large –The result is a residual plot that shows a curved pattern Conclusion: If the residual plot looks like a patternless “cloud of points” then the LSRL is a good model of the data

A Calculator Trick In your STAT : EDIT screen, scroll right so you can see L 3 and L 4 Paste a list called RESID into L 4 –It’s in your calculator already! Note that RESID is identical to L 3 Every time you run LinReg, the calculator computes all the residuals and places them in RESID Edit Stat Plot 3 so that the Ylist is based on RESID instead of L 3 –Now leave it that way! –After every LinReg, Plot 3 will always show you the residual plot.

The Sanchez Residual Plot Paste the Sanchez gas data GASDA & GASFT into L 1 & L 2 Run LinReg L 1, L 2, Y 1 Turn off Plot 3 and turn on Plot 1 –Tap Zoom-9 to see the graph Turn off Plot 1 and Y 1 and turn on Plot 3 –Tap Zoom-9 to see the residual plot Sketch the residual plot in your notes Write a sentence that describes what the residual plot tells you about how well the LSRL fits the data.

Fathom Demo Look at Sanchez data in Fathom Add moveable line, show squares, show residuals Adjust slope and intercept of moveable line and observe effect on residual plot

Objectives Calculate the value of a residual at a point on a LSRL Create a residual plot on the calculator and draw a conclusion about the goodness of fit of the LSRL

Homework Read pages 151 – 159 Do problems 39, 40, 41