1. Baseline Regression Fit.
Does the regression line appear to do a reasonable job of summarizing the overall linear relationship in these data observations?
Solution.
Opinions may vary.
Section 26.5 Applet Exploration: Behavior of Regression Lines
Exercises Exploration Tasks
Recall the 20 observations are height and foot length for a sample of statistics students. Use the Analyzing Two Quantitative Variables applet. The data have been preloaded into the applet.
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Check the Show Regression Line box.
2. Predict Effect of New Point.
If we were to add an observation with height 60 in and foot length 35 cm, predict whether, and if so how, the regression line will change.
Solution.
Should expect this observation to “pull the line down” (smaller slope).
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Check Show data options and enter these coordinates (\(x = 35\text{,}\) \(y = 60\)) to add an observation. Press the Add button.
3. Add Point and Report New Equation.
Report the new regression equation.
Solution.
Regression equation will vary.
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Check the Move observations box. Click on this new observation (it should change color) and hold the mouse button down and move the mouse vertically in both directions to change the \(y\) value of the observation (try hard not to change the \(x\) value). The applet automatically recalculates the new regression line depending on the new location of the point.
4. Move New Point Vertically.
Is it possible to make the regression line have a negative slope? Does the regression line appear to be affected by the location of this point? Is the impact strong or weak? Does this match your prediction?
Solution.
If you pull down far enough (e.g., height < 20) then you can get the regression line to have a negative slope. This point does appear to have moderate impact on the regression line.
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Delete this new point and/or press the Revert button to return to the original data set.
5. Predict Effect of Central Point Movement.
Now focus on the point located at (29, 65). If we move this point vertically, predict how the regression line will change. Do you think the change will be as dramatic as in Question 3?
Solution.
predictions will vary.
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Repeat Question 4 using the point located at (29, 65).
6. Move Central Point.
Does the regression line appear to be affected by the location of this point? Is the impact strong or weak (especially compared to the impact you witnessed in Question 4)?
Solution.
The regression line does not appear to be nearly as affected by changes in the \(y\)-coordinate for this observation.
7. Influence Comparison.
Which point was more influential on the equation of the regression line: (35, 60) or (29, 65)? Suggest an explanation for why the point you identified is more influential, keeping in mind the “least-squares criterion.”
Solution.
(35, 60), because it has a more extreme (far from the average) \(x\)-coordinate.
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