Question

Cars lose value the farther they are driven. A random sample of \[11\] cars for sale was taken. All \[11\] cars were the same make and model. A line was fit to the data to model the relationship between how far each car had been driven and its selling price. A scatterplot plots points x y axis. The y axis is labeled Price in thousands of dollars. The x axis is labeled Kilometers driven in thousands. Points fall diagonally in a relatively narrow pattern. The line passes between the points through (0, 40) and (100, 15). All values estimated. \[\small{10}\] \[\small{20}\] \[\small{30}\] \[\small{40}\] \[\small{50}\] \[\small{60}\] \[\small{70}\] \[\small{80}\] \[\small{90}\] \[\small{100}\] \[\small{5}\] \[\small{10}\] \[\small{15}\] \[\small{20}\] \[\small{25}\] \[\small{30}\] \[\small{35}\] \[\small{40}\] A scatterplot plots points x y axis. The y axis is labeled Price in thousands of dollars. The x axis is labeled Kilometers driven in thousands. Points fall diagonally in a relatively narrow pattern. The line passes between the points through (0, 40) and (100, 15). All values estimated. Which of these linear equations best describes the given model? Choose 1 answer: Choose 1 answer: (Choice A) \[\hat y=\dfrac{1}2x+40\] A \[\hat y=\dfrac{1}2x+40\] (Choice B) \[\hat y=-x+40\] B \[\hat y=-x+40\] (Choice C) \[\hat y=-\dfrac{1}4x+40\] C \[\hat y=-\dfrac{1}4x+40\] Based on this equation, estimate the price of a car that had been driven \[56\] thousand kilometers. \[\$\] thousand dollars

Answer 1

The correct equation is ŷ = -¼x + 40 (Choice C). To estimate the price for a car driven 56 thousand kilometers, the equation predicts a price of $26,000.

Step-by-step explanation:

The question is asking to determine which linear equation best describes the relationship between the kilometers driven in thousands (x-axis) and the price in thousands of dollars (y-axis) for a sample of cars, and then to use that equation to estimate the price of a car that has been driven 56 thousand kilometers.

The line passes through the points (0, 40) and (100, 15), which can be used to find the slope of the line, m, using the formula m = (y2 - y1) / (x2 - x1). The slope is (15 - 40) / (100 - 0) = -25 / 100 = -0.25. Therefore, the equation of the line must have this slope. The y-intercept is given by the point (0, 40), which means the correct equation is Price = -0.25(Kilometers) + 40, or in the given format ŷ = -¼x + 40, which corresponds to Choice C.

To estimate the price of a car that has been driven 56 thousand kilometers, we substitute x = 56 into the regression equation to get ŷ = -0.25(56) + 40 = -14 + 40 = 26. Hence, the estimated price is $26,000.