Question

In the United States, tyre tread depth is measured in 32nds of an inch. Car tyres typically start out with 10/32 to 11/32 of an inch of tread depth. In most states, a tyre is legally worn out when its tread depth reaches 2/32 of an inch. A random sample of four tyres provides the following data on distance driven and tread depth: Distance driven Tread Depth (32nds of an inch) Tyre (10,000 kilometres) 2 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. TREAD DEPTH (32nds of an) 10 Sum of Distances [x-bar, y-bar) 3 MILEAGE (10,000 kilometres) Help Clear All Think about how close the line y-9 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 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 (x, ỹ), wherex is the mean distance driven for the four tyres in the sample and у is the mean tread depth for the four tyres in the sample. The line you just plotted through the point (X,-). does not pass 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 O The line you plotted that has a sum of the distances equal to0 Now think about the population of tyres. Each tyre in the population has a value of x (its distance driven) 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 variable is: You can use sample data to estimate the parameters βο and β1 and obtain the following estimated regression equation, where y is the predicted value of y: ybo+ bix The difference between y and y for a particular sample point (observation) is called a residual. Imagine 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 b 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 = ỹ line. There are three sums of squares that are foundational to regression analysis: the SS(Total) (variation in y), SSR (sum of squares for regression), and SSE (sum of squares for error). Regardless of the sum of squares, for each observation, a particular distance is measured and squared, and then the squares are summed Consider tyre 1. Use the green points (triangle symbol) to mark the distance for tyre 1 that is squared and included in the SSR. Then use the purple points (diamond symbol) to mark the distance for tyre 1 that is squared and included in the SSE. Line segments will automatically connect the points. TREAD DEPTH (32nds of an inch) 10 Least Squares Line yay-bar SSR Building Block SSE Building Block 3 LHelp TYRE MILEAGE 110,000 kilometres) ClearAll The value of the SSE is 2 0 7.5 ONA 3 16 2004-2016 Anlia

Answer 1

9-2x : 7, 5, 3, 1 (y-j)=0+0+1+3=4 The sum of the vertical distances between the sample points and line is 4, and the sum of the squared vertical distances between the sample points and the line is 02 + 02 12+32-10 r (1+2+3+4)/4 2.5. У-(7+5+4+4)/4-5 The line you just plotted passes through the point (T, y) The sum of the squared vertical distances between the sample points and the line you just plotted is 02+021232 10 Neither - the two lines fit the data equally well you know that the sum of the residuals is zero. In addition, you know that the sum of squared residuals is minimum bị = (45-(10 * 20)/4)/(30_ (10)2/4) =-5/5 =-1 y intercept =--bix = 7,5: where, x =- Vi=5 ss(Total) = (Vi-Vi)2-1 where yi-7.5-x SSE- SSR-SST- SSE-5 ns SSE 1