Chapter 4.2, Problem 19P

Expand Your Knowledge: Residual Plot The least-squares line usually does not go through all the sample data points (x, y). In fact, for a specified x value from a data pair (x, y), there is usually a difference between the predicted value and the y value paired with x. This difference is called the residual.

The residual is the difference between the y value in a specified data pair (x, y) and the value $\hat{y}=a+bx$ predicted by the least-squares line for the same x.

$y-\hat{y}$ is the residual.

One way to assess how well a least-squares line serves as a model for the data is a residual plot. To make a residual plot, we pull the x values in order on the horizontal axis and plot the corresponding residuals $y-\hat{y}$ in the vertical direction. Because the mean of the residuals is always zero for a least-squares model, we place a horizontal line at zero. The accompanying figure shows a residual plot for the data of Guided Exercise 4, in which the relationship between the number of ads run per week and the number of cars sold that week was explored. To make the residual plot, first compute all the residuals. Remember that x and y are the given data values and that $\hat{y}$ is computed from the least-squares line $\hat{y}\approx 6.56+1.01x$.

Residual
X	y		$\hat{y}$		$y-\hat{y}$
6	15		12.6		2.4
20	31		26.8		4.2
0	10		6.6		3.4
14	16		20.7		-4.7
25	28		31.8		-3.8
Residual
X		y		$\hat{y}$		$y-\hat{y}$
16		20		22.7		-2.7
28		40		34.8		5.2
18		25		24.7		0.3
10		12		16.7		-4.7
8		15		14.6		0.4

If the least-squares line provides a reasonable model for the data, the pattern of points in the plot will seem random and unstructured about the horizontal line at 0. Is this the case for the residual plot?

If a point on the residual plot seems far outside the pattern of other points, it might reflect an unusual data point (x. y), called an outlier. Such points may have quite an influence on the least-squares model. Do there appear to be any outliers in the data for the residual plot?