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3.4 Two-Variable Data & Models - Line of best fit
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The figure presents a sterplot in the first quadrant of the x y plane. The numbers 0 through 10, in increments of 2, are indied on the x axis. The numbers 0 through 30, in increments of 5, are indied on the y axis. There are 8 data points in the sterplot. 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 sterplot above?

A.

y equals, 3 x plus 0 point 8

B.

y equals, 0 point 8 x, plus 3

C.

y equals, negative 0 point 8 x, plus 3

D.

y equals, negative 3 x plus 0 point 8

Which of the following could be the equation for a line of best fit for the

Hard-difficulty · SAT Math · Two-Variable Data & Models — Line of best fit. Read the question above, select your answer, and check the full explanation below to understand exactly why the correct choice works.

Answer 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.

 

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