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Applet Exercises

Applet Exercises

To do these exercises, go to www.macmillanhighered.com/fapp10e.

Question 6.96

1. In the Correlation and Regression applet, imitate Figure 6.16 (page 269). Click to locate five points at the lower left of the scatterplot, and then click “Show least- squares line.”

  1. What is the correlation r for these five points? If necessary, move points with the mouse to get a value near r=0.5, as in Example 9 (page 268).
  2. Now add an outlier at the upper right that lies exactly on the line. What is the correlation r for the six points?
  3. Use the mouse to drag the outlier down and then to the left. Watch the least-squares line follow this one point. How negative can you make the correlation r?
  4. What have you learned from parts (a) through (c) about the effect that a single outlier can have on the correlation r?

Question 6.97

2. You are going to use the Correlation and Regression applet to make different scatterplots with 10 points that have a correlation close to 0.7. Many patterns can have the same correlation. Always plot your data before you trust a correlation.

  1. Stop after adding the first two points. What is the value of the correlation? Why does it have this value no matter where the two points are located?
  2. Make a lower-left to upper-right pattern of 10 points with a correlation of about r=0.7. (You can drag points up or down to adjust r after you have 10 points.) Make a rough sketch of your scatterplot.
  3. Make another scatterplot with nine points in a vertical stack at the left of the plot. Add one point far to the right and move it until the correlation is close to 0.7. Make a rough sketch of your scatterplot.
  4. Make yet another scatterplot with 10 points in a curved pattern that starts at the lower left, rises to the right, then falls again at the far right. Adjust the points up or down until you have a smooth curve with a correlation close to 0.7. Make a rough sketch of this scatterplot as well.
  5. Based on your answers to parts (b) through (d), what can you conclude about the pattern of dots in a scatterplot if r≈0.7?

Question 6.98

3. It isn’t easy to guess the position of the least-squares line by eye (at least not without some practice). Use the Correlation and Regression applet to compare a line that you draw with the least-squares line. Click on the scatterplot to create a group of 15 to 20 points from the lower left to the upper right with a clear, positive straight-line pattern (with a correlation of around 0.7). Click the “Draw your own line” button. Then click on two points, one in the lower left and the other in the upper right. Move these two points so that your line is drawn through the middle of the cloud of points and appears to do a good job of summarizing the pattern in the scatterplot.

Page 289
  1. You drew a line by eye through the middle of the pattern. Read off the value for Relative SS (directly below the “Draw your own line” button). This is the ratio of the sum of squares of residuals for your line and the sum of squares of the residuals for the least-squares regression line.
  2. Move your line until the Relative SS=1. (This means that the value of Relative SS is approximately 1, or equals 1 when rounded to two decimal places.) Your line should now closely match that of the least-squares regression line. To check this out, click the “Show least-squares line” box.
  3. Repeat this exercise several times with different sets of points. Try to guess the “best” line without looking at the value of Relative SS. Does your ability to pick a line that closely matches the least-squares regression line improve with practice?

Question 6.99

4. This time you will use the Correlation and Regression applet to examine the residual errors. Click Clear to remove any work done for Exercises 1–3. Click on the scatterplot to create a group of 15 to 20 points from the lower left to the upper right with a clear, positive straight-line pattern (with a correlation in the moderate range, say between 0.65 and 0.75). Click the “Show least-squares line” and “Show residuals” buttons.

  1. How can you tell from this graph which of the residuals are positive and which are negative? Do the residuals appear roughly balanced between positive and negative values?
  2. Pick a point that has an x-value somewhere in the middle of x-values of the other data points. If this point lies below the line, drag it vertically down. If this point lies above the line, drag it up. As you drag the point vertically, what happens to the size of its residual? What happens to the slope (or the tilt) of the least- squares line? Then switch the direction in which you drag the point and note the effect on the slope of the line.
  3. Return the point you were dragging in part (b) to approximately its original position. Next, click on a point that has the largest x-value. Try dragging this point vertically both in the upward and downward direction. What effect does this have on the slope of the least-squares line?
  4. Which type of outlier, one with an x-value that lies near the middle of the x-values of the other data points or one with an x-value that lies near the maximum (or minimum) of the x-values of the other data points, will have a greater influence on the slope of the least-squares line?
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