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Problem-Solving and Data Analysis Difficulty: Hard

The figure presents a scatterplot titled “Size and Sale Price of Houses in Town H.” The x axis is labeled “Size, in thousands of square feet,” and the numbers 0 through 3, in increments of 1, are indicated. The y axis is labeled “Sale price, in thousands of dollars,” and the numbers 0 through 400, in increments of 50, are indicated.

There are 25 data points on the scatterplot. The data points begin in the middle left part of the graph, at the point with coordinates zero point 9 comma one hundred ninety. They trend upward and to the right in clusters, ending at the point with coordinates 2.55 comma 350.

A line of best fit for the data is not shown in the graph. However, it appears that if a line of best fit were drawn, it would pass through the point with coordinates zero comma one hundred and the point with coordinates 2 point 5 comma three hundred fifty.

The scatterplot above shows the size x and the sale price y of 25 houses for sale in Town H. Which of the following could be an equation for a line of best fit for the data?

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Explanation

Choice B is correct. From the shape of the cluster of points, the line of best fit should pass roughly through the points $with coordinates 1 comma 200$ and $2 point 5 comma 350$ . Therefore, these two points can be used to find an approximate equation for the line of best fit. The slope of this line of best fit is therefore $the fraction with numerator y sub 2, minus y sub 1, and denominator x sub 2, minus x sub 1, end fraction, equals, the fraction with numerator 350 minus 200, and denominator 2 point 5 minus 1, end fraction$ , or 100. The equation for the line of best fit, in slope-intercept form, is $y equals, 100 x plus b$ for some value of b. Using the point $with coordinates 1 comma 200$ , 1 can be substituted for x and 200 can be substituted for y: $200 equals, 100 times 1, plus b$ , or $b equals 100$ . Substituting this value into the slope-intercept form of the equation gives $y equals, 100 x plus 100$ .

Choice A is incorrect. The line defined by $y equals, 200 x plus 100$ passes through the points $with coordinates 1 comma 300$ and $2 comma 500$ , both of which are well above the cluster of points, so it cannot be a line of best fit. Choice C is incorrect. The line defined by $y equals, 50 x plus 100$ passes through the points $with coordinates 1 comma 150$ and $2 comma 200$ , both of which lie at the bottom of the cluster of points, so it cannot be a line of best fit. Choice D is incorrect and may result from correctly calculating the slope of a line of best fit but incorrectly assuming the y-intercept is at $the point with coordinates 0 comma 0$ .