In the United States, tire tread depth is measured in 32nds of an inch. Car tires typically start out with 10/32 to 11/32 of an inch of tread depth. In most states, a tire is legally worn out when its tread depth reaches 2/32 of an inch.
A random sample of four tires provides the following data on mileage and tread depth:
Think about how close the line y = 9 – 2 x is to the sample points. Look at the graph and find each point’s vertical distance from the line. If the point sits above the line, the distance is positive; if the point sits below the line, the distance is negative.
The sum of the vertical distances between the sample points and line is A) 5, B) 4, C)2.5, D) 0, and the sum of the squared vertical distances between the sample points and the line is A) 0, B) 10, C)2.5, D) 1.
Use the green line (triangle symbols) to plot the line on the graph that has the same slope as the line y = 9 – 2x, but with the additional property that the vertical distances between the points and the line sum to 0. Then use the black point (X symbol) to plot the point (x̄ , ȳ ), where x̄ is the mean mileage for the four tires in the sample and ȳ is the mean tread depth for the four tires in the sample.
The line you just plotted A) does not pass, B) passes, through the point (x̄ , ȳ ).
The sum of the squared vertical distances between the sample points and the line you just plotted is A) 10, B) 1, C) 6, D) 0.
According to the criterion used in the least squares method, which of the two lines provides a better fit to the data?
A) The line you plotted that has a sum of the distances equal to 0
B) y = 9 – 2x
C) Neither—the two lines fit the data equally well
Suppose you fit a least squares regression line to the four sample points on the graph. Based on your work so far, even before you fit the line, you know that the sum of the residuals is A) between 0 and 1, B) 0, C) less than 0, D) between -1 and 1. In addition, being as specific as you can be, you know that the sum of the squared residuals is A) between 6 and 10, B) 0, C) less than or equal to 6, D) less than equal to 10.
The y intercept of the least squares regression line is A) 3.5, B) 7.5, C) 5, D) 4.5.
The value of the SSE is A) 2, B) 0, C) 7.5, D) 1.
