Question

Data on the fuel consumption ?y of a car at various speeds ?x is given. Fuel consumption is measured in mpg, and speed is measured in miles per hour. Software tells us that the equation of the least‑squares regression line is

?̂ =55.3286−0.02286?y^=55.3286−0.02286x

Using this equation, we can add the residuals to the original data.

Speed	1010	2020	3030	4040	5050	6060	7070	8080
Fuel	38.138.1	54.054.0	68.468.4	63.663.6	60.560.5	55.455.4	50.650.6	43.843.8
Residual	−17.00−17.00	−0.87−0.87	13.7613.76	9.199.19	6.316.31	1.441.44	−3.13−3.13	−9.70−9.70

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(a) Use the software of your choice to make a scatterplot of the observations. Have the software include the regression line on the plot, or print the scatterplot and draw the regression line provided on your plot.

(b) Would you use the regression line to predict ?y from ??x?

No, the pattern is nonlinear, so linear regression is not appropriate for prediction.

No, the pattern is a straight line, so linear regression is not appropriate for prediction.

Yes, the pattern is nonlinear, so linear regression is appropriate for prediction.

Yes, the pattern is a straight line, so linear regression is appropriate for prediction.

(c) On a separate sheet of paper, verify the value of the first residual, for ?=10.x=10. Then verify that the residuals add up to 0.0.

(d) Now use the residuals and ?-valuesx-values in the table to make a plot of the residuals against the values of ?,x, also on a separate sheet of paper. Draw a horizontal line at height 00 on your plot.

Compare your scatter plot with your regression plot. How does the pattern of the residuals about this line compare with the pattern of the data points about the regression line in your original scatterplot?

The observations in the scatter plot that are above the regression line have

and the observations in the scatter plot that are below the regression line have

The distance of the observations above the regression line is

the value of the residual, and the distance of the observations below the regression line is

the absolute value of the residual. Therefore, the pattern of the residuals is

the pattern of the data points about the regression line in the original scatterplot.

Answer 1

(a)

(b) No, the pattern is nonlinear, so linear regression is not appropriate for prediction.

(c)

The regression equation is
Fuel = 55.329 - 0.023 Speed

Residual for x=10 =(38.1-55.329 - 0.023 *10)=-17.459

(d)

Residuals: -16.999, -0.869, 13.761, 9.191, 6.321, 1.451, -3.119, -9.689

The distance of the observations above the regression line is the value of the residual, and the distance of the observations below the regression line is the absolute value of the residual. Therefore, the pattern of the residuals is the pattern of the data points about the regression line in the original scatter plot.