## Linear Regression I

*regression line*, sometimes known as a "line of best fit."

Plot several points in a relatively straight line, and then click **Show Line**. How well does the line approximate the scatterplot? Then, plot another point that does not lie along the line. How does the regression line change because of the outlier?

*Ctrl*key, and click on the point you wish to delete. To move a point, hold down the

*Shift*key, and drag the point to a new location with the mouse.

To change the scale of the graph, change the values of x‑min, x‑max, y‑min, and y‑max, and hit the **Set Scale** button.

Click **Show Line** to display the linear regression line.

In the upper left corner, the following values are displayed:

n | The number of points on the graph. |

r | The correlation coefficient. This measure indicates the association between the x‑variable and the y‑variable. Its absolute value roughly indicates how well the line of best fit approximates the data. |

y = | An equation describing the line of best fit. |

*r*-value and the equation of the line only appear after

**Show Line**is clicked.)

The **Clear** button will reset the graph.

Plot one point on the graph and then click **Show Line**. Why do you think a line is not graphed?

**Clear** the graph and plot two points that have whole-number coordinates.

- On your own, find an equation for the line through these two points.
- Click
**Show Line**. Compare the equation for the line drawn to the equation that you calculated. Explain and resolve any differences.

**Clear** the graph and plot three points. Think about a line that "fits" these three points as closely as possible.

- Is it possible for a single straight line to contain all three of the points you plotted?
- On a piece of paper, plot these same three points, and sketch a line that you think best fits the three points.
- Click
**Show Line**. Do you think that the line graphed fits the points well? How does it compare to the line you drew?

**Clear** the graph. Place several points on the graph that lie roughly in a straight line, then hit **Show Line**. The line that appears is the *regression line*, which is sometimes known as the "line of best fit."

- What is the
*r*-value for the line? - Place just one additional point on the graph that lies far away from the line. What effect does this point have on the
*r*‑value? What effect does it have on the line of best fit? - Move several of the points. How does the
*r*-value and line change as points are moved?

The line that is drawn is called the "least-squares regression line." Bascially, the least-squares regression line is the line that minimizes the squared "errors" between the actual points and the points on the line. This makes the line fit the points. To get a better feel for the regression line, try the following tasks.

- Plot four points so that the regression line is horizontal. Do this in several different ways. What do you notice about the regression line and the
*r*‑value? - Plot three points (not all on a straight line) so that the regression line is horizontal. What do you notice about the regression line and the
*r*‑value?