tn the United States, tire tread depth is measured in 32nds of an inch. Car tire

Question

tn the United States, tire tread depth is measured in 32nds of an inch. Car tires typically start out with 10/32 to 11/32 o an inch of treed depth. In most states, a tire is legally worn out when its tread depth reaches 2/32 of an lnch A random sample of four tires provides the following data on mileage and tread depth: Mileage Tread Depth Tire (10,000 miles) (32nds of an inch) A scatter diagram of the sample data is shown below (blue points). The line y 11 - 2x is also shown in orange TREAD DEPTH (32nds of an 10 1 Sure of Dinstances x-bar,y-bar Think about how close the line y 11- 2x 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 ine, 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 distances between the sample points and the line is , and the sum of the squared vertical Use the green line (triangle symbols) to plot the line on the graph that has the same slope as the line y -11 -2x, but with the additional property that the vertical distances between the points and the ine sum to 0. Then use the black point (x symbol) to plot the point (R. 9), whereisthe mean mileage for the four tires in the sample and is the mean tread depth for the four tires in the sample. The line you just plotted through the point (R,9) and the line you just plotted is The sum of the squared vertical distances between the sample points According to the citerion used in the least squares method, which of the two lines provides a better fit to the data? O Neither-the two lines fit the data equally wel O The ine you plotted that has a sum of the distances equal to 0

Explanation / Answer

The sum of the vertical distences between the sample points and line is (-1)+ (1) + 2 + 2 = 4

and the sum of square vetical distances are 12 + 12 + 22 +22 = 10

Here x = (1 + 2 + 3 + 4)/4 = 2.5 and ybar = (8 + 8 + 7 + 5)/4 = 7

so the line y = a -2x passes through (2,5,7)

so a = y + 2x = 7 + 2.5 * 2 = 12

so predcited value, sum of error and sum of square error is given below.

So, THe line we just plotted is y = 12 - 2x passes through (2.5,7)

THe sum of squared vertical distances between the sample points and the line we just plotted is 6.

Here obviouly,

the line we plotted that has a sum of the distaces equal to Zero would be a better fit. Option b is correct.

Suppose we fit a linear regression line for the four given points, we know the sum of the residuals is 0.You know the sum of the squared residuals is less than 6 or say 1.

b1 = SSXY /SSXX

SSXY =  (xy) - (x)((y)]/n = 65 - 10 * 28/4 = -5

SSXX = (x2 ) - (x)2/n   = 30 - 102/4 = 5

b1 = (-5/5) = -1

as least square regression line y^ = bo + b1 x will pass throught the (x, ybar) or (2.5,7)

b0 = 7 - 2.5 * (-1) = 9.5

so least square regression line is

y^ = 9.5 - x

SSE = 1

X Y Y' = 12 - 2X Y-Y' (Y - Y')^2 1 8 10 -2 4 2 8 8 0 0 3 7 6 1 1 4 5 4 1 1 Sum 10 28 28 0 6 Average 2.5 7

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