2.109 Use the applet. It isn’t easy to guess the position of the least-squares line by eye. Use the Correlation and Regression applet to compare a line you draw with the least-squares line. Click on the scatterplot to create a group of 15 points from lower left to upper right with a clear, positive straight-line pattern (correlation around 0.6). Click the “Draw line” button and use the mouse to draw a line through the middle of the cloud of points from lower left to upper right. Note the “thermometer” that appears above the plot. The red portion is the sum of the squared vertical distances from the points in the plot to the least-squares line. The green portion is the “extra” sum of squares for your line—it shows by how much your line misses the smallest possible sum of squares.

1. (a) You drew a line by eye through the middle of the pattern. Yet the right-hand part of the bar is probably almost entirely green. What does that tell you?

2. 

   (b) Now click the “Show least-squares line” box. Is the slope of the least-squares line smaller (the new line is less steep) or larger (line is steeper) than that of your line? If you repeat this exercise several times, you will consistently get the same result. The least-squares line minimizes the vertical distances of the points from the line. It is not the line through the “middle” of the cloud of points. This is one reason it is hard to draw a good regression line by eye.