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Problem-Solving and Data Analysis / Two-variable data: Models and scatterplots Difficulty: Medium

The figure presents a scatterplot in the first quadrant of the x y plane. The numbers 0 through 10, in increments of 2, are indicated on the x axis. The numbers 0 through 30, in increments of 5, are indicated on the y axis. There are 8 data points in the scatterplot. The data points begin at the point with approximate coordinates 1 comma 4, and trend upward and to the right until they end at the point with approximate coordinates 8 comma 26.

Which of the following could be the equation for a line of best fit for the data shown in the scatterplot above?

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Explanation

Choice A is correct. The data show a strong linear relationship between x and y. The line of best fit for a set of data is a linear equation that minimizes the distances from the data points to the line. An equation for the line of best fit can be written in slope-intercept form, $y equals, m x plus b$ , where m is the slope of the graph of the line and b is the y-coordinate of the y-intercept of the graph. Since, for the data shown, the y-values increase as the x-values increase, the slope of a line of best fit must be positive. The data shown lie almost in a line, so the slope can be roughly estimated using the formula for slope, $m equals, the fraction with numerator y sub 2 minus y sub 1, and denominator x sub 2 minus x sub 1, end fraction$ . The leftmost and rightmost data points have coordinates of about $1 comma 4$ and $8 comma 26$ , so the slope is approximately $the fraction with numerator 26 minus 4, and denominator 8 minus 1, end fraction, equals the fraction 22 over 7$ , which is a little greater than 3. Extension of the line to the left would intersect the y-axis at about $the point with coordinates 0 comma 1$ . Only choice A represents a line with a slope close to 3 and a y-intercept close to $the point with coordinates 0 comma 1$ .

Choice B is incorrect and may result from switching the slope and y-intercept. The line with a y-intercept of $0 comma 3$ and a slope of 0.8 is farther from the data points than the line with a slope of 3 and a y-intercept of $0 comma 0 point 8$ . Choices C and D are incorrect. They represent lines with negative slopes, not positive slopes.