Least-Squares Property According to the least-squares property, the regression line minimizes the sum of the squares of the residuals. Refer to the jackpot/tickets data in Table 10-1 and use the regression equation y^ = -10.9 + 0.174x that was found in Examples 1 and 2 of this section.
b. Find the sum of the squares of the residuals.

Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:

y^ = 58.9 - 0.00749x, where x represents weight.

c. What is the predictor variable?

Notation The author conducted an experiment in which the height of each student was measured in centimeters and those heights were matched with the same students’ scores on the first statistics test.

c. Does r change if the heights are converted from centimeters to inches?

Variation and Prediction Intervals

In Exercises 17–20, find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. In each case, there is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions.

Altitude and Temperature Listed below are altitudes (thousands of feet) and outside air temperatures (°F) recorded by the author during Delta Flight 1053 from New Orleans to Atlanta. For the prediction interval, use a 95% confidence level with the altitude of 6327 ft (or 6.327 thousand feet).

Sum of Squares Criterion In addition to the value of another measurement used to assess the quality of a model is the sum of squares of the residuals. Recall from Section 10-2 that a residual is (the difference between an observed y value and the value predicted from the model). Better models have smaller sums of squares. Refer to the U.S. population data in Table 10-7.

c. Verify that according to the sum of squares criterion, the quadratic model is better than the linear model.

Outlier Refer to the accompanying Minitab-generated scatterplot.

b. After identifying the 10 pairs of coordinates corresponding to the 10 points, find the value of the correlation coefficient r and determine whether there is a linear correlation.

Notation The author conducted an experiment in which the height of each student was measured in centimeters and those heights were matched with the same students’ scores on the first statistics test.

b. Without doing any research or calculations, estimate the value of r.