(Solved): 1. First-order linear model In the United States, tire tread depth is measured in 32nds of an inch. ...

1. First-order linear model 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: A scatter diagram of the sample data follows [blue points (circle symbols)]. The line $y=6-x$ is also shown in orange. Think about how close the line $y=6-\mathrm{x}$ 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 betwee sample points and the orange line is , and the sum of the squared vertical distances between the sample points and the line is On the graph, use the green line (triangle symbols) to plot the line that has the same slope as the line $y=6-x$, but with the additional property that the vertical distances between the points and the line sum to 0 . Then place the black point (X symbol) on the graph to plot the point ( $\overline{\mathrm{x}},\overline{\mathrm{y}}$ ) where $\overline{\mathrm{x}}$ is the mean mileage for the four tires in the sample and $\overline{\mathrm{y}}$ is the mean tread depth for the four tires in the sample. The line you just plotted through the point $(\bar{x},\bar{y})$. The sum of the squared vertical distances between the sample points and the line that 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? The line you plotted that has a sum of the distances equal to 0 Neither-the two lines fit the data equally well $y=6-x$ 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 some error variable is: $y={\beta }_0+{\beta }_{1}x+\epsilon$ You can use the sample data to estimate the parameters ${\beta }_0$ and ${\beta }_1$ and obtain the following estimated regression equation, where $\hat{y}$ is the predicted value of $y$ : $\hat{y}=b_0+b_{1}x$ The difference between $\mathrm{y}$ and $\hat{\mathrm{y}}$ for a particular sample point (observation) is called a residual. Suppose you use the least squares method to find the least squares line for the four sample points on the graph. On the basis of your work so far, even before you fit the line, you know that the sum of the residuals is . In addition, you know that the sum of the squared residuals is An alternate formula for the slope of the least squares line follows. Enter the values for the numerator and denominator, and then enter the 1. ${\mathrm{b}}_1=\frac{\sum {\mathrm{x}}_{\mathrm{i}}{\mathrm{y}}_{\mathrm{i}}-\left(\sum {\mathrm{x}}_{\mathrm{i}}\sum {\mathrm{y}}_{\mathrm{i}}\right)/\mathrm{n}}{\sum {\mathrm{x}}_{\mathrm{i}}^2-{\left(\sum {\mathrm{x}}_{\mathrm{i}}\right)}^2/\mathrm{n}}=$ The $y$-intercept of the least squares line is On the following scatter diagram of the blue sample points (circle symbols), use the orange line (square symbols) to plot the least squares line. Place the first orange square at the left edge of the graph (the $y$-intercept) and the second orange square at the value of $\hat{y}$ at the right edge of the graph.