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 1 2 3 4 Mileage (10,000 miles) 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: b = y = Bo + Bx + 1 2 3 4 You can use sample data to estimate the parameters Bo and B and obtain the following estimated regression equation, where is the predicted value of y: = Tread Depth (32nds of an inch) 7 6 4 1 bo + bix The difference between y and y for a particular data point (observation) is called a residual. Exiyi (Exiyi) / n Ex - (Exi)/n The formula for the slope of the least-squares line is provided as follows. Enter the values for the numerator and denominator, and then enter the value of b. The sample mean x for mileage is the least-squares line is TREAD DEPTH (32nds of an inch) On the scatter diagram that follows, use the orange line (square symbol) to plot the least-squares regression line, and then use the olive line (dash symbol) to plot the y = y line. 10 9 8 7 6 5 4 3 2 1 0 O 1 O , and the sample mean y for tread depth is O Least Squares Line 0 y=y-bar SSR Building Block The value of the SSE (for all the data points) is SSE Building Block 2 3 4 5 TIRE MILEAGE (10,000 miles) Help Clear All . Thus, the y-intercept of Three sums of squares 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 4. Use the green points (triangle symbol) to mark the distance for tire 4 that is squared and included in the SSR. Then use the purple points (diamond symbol) to mark the distance for tire 4 that is squared and included in the SSE. Line segments will automatically connect the points.