1. The least squares criterion, SSE, SSR, and SST
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:
Tire 
Mileage 
Tread Depth 

(10,000 miles) 
(32nds of an inch) 

1  1  7 
2  2  5 
3  3  4 
4  4  4 
A scatter diagram of the sample data is shown below (blue points). The line y = 9 – 2x is also shown in orange.
0 Sum of Distances(xbar, ybar)0123451086420TREAD DEPTH (32nds of an)MILEAGE (10,000 miles)
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 , and the sum of the squared vertical distances between the sample points and the line is .
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 (x̄x̄, ȳȳ), where x̄x̄ is the mean mileage for the four tires in the sample and ȳȳ is the mean tread depth for the four tires in the sample.
The line you just plotted through the point (x̄x̄, ȳȳ).
The sum of the squared vertical distances between the sample points and the line you just plotted is .
According to the criterion used in the least squares method, which of the two lines provides a better fit to the data?
Neither—the two lines fit the data equally well
The line you plotted that has a sum of the distances equal to 0
y = 9 – 2x
Now think about the population of tires. Each tire in the population has a value of x (its mileage) and a corresponding value of y (its tread depth). If the relationship between x and y is linear, the equation that describes how y is related to x and an error term is:
y = β00 + β11x + ε
You can use sample data to estimate the parameters β00 and β11 and obtain the following estimated regression equation, where ŷ is the predicted value of y:
ŷ = b00 + b11x
The difference between y and ŷ for a particular sample point (observation) is called a residual.
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 . In addition, being as specific as you can be, you know that the sum of the squared residuals is .
An alternate formula for the slope of the least squares line is provided below. Enter the values for the numerator and denominator, and then enter the value of b1b1.
b1b1  = =  ∑xiyi − (∑xiyi)/n∑xi2 − (∑xi)2/ n∑xiyi − ∑xiyi/n∑xi2 − ∑xi2/ n  = =  / /  = = 
The y intercept of the least squares regression line is .
On the below scatter diagram of the sample points, use the orange line (square symbols) to plot the least squares regression line, then use the olive line (dash symbols) to plot the y = ȳȳ line.
There are three sums of squares that are foundational to regression analysis: the SST (total sum of the squares), SSR (sum of squares due to regression), and SSE (sum of squares due to error). Regardless of the sum of squares, for each observation, a particular distance is measured and squared, and then the squares are summed.
Consider tire 1. Use the green points (triangle symbol) to mark the distance for tire 1 that is squared and included in the SSR. Then use the purple points (diamond symbol) to mark the distance for tire 1 that is squared and included in the SSE. Line segments will automatically connect the points.
Least Squares LinesSSR Building BlockSSE Building Blocky = ybar012345109876543210TREAD DEPTH (32nds of an inch)TIRE MILEAGE (10,000 miles)
The value of the SSE is .
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