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Problem-Solving and Data Analysis Difficulty: Hard

The scatterplot shows the relationship between two variables, $x$ and $y$ , for data set E. A line of best fit is shown. Data set F is created by multiplying the y-coordinate of each data point from data set E by $3.9$ . Which of the following could be an equation of a line of best fit for data set F?

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Explanation

Choice A is correct. An equation of a line of best fit for data set F can be written in the form $y = a + b x$ , where $a$ is the y-coordinate of the y-intercept of the line of best fit and $b$ is the slope. The line of best fit shown for data set E has a y-intercept at approximately $left parenthesis 0 comma 12 right parenthesis$ . It's given that data set F is created by multiplying the y-coordinate of each data point from data set E by $3.9$ . It follows that a line of best fit for data set F has a y-intercept at approximately $left parenthesis 0 comma 12 left parenthesis 3.9 right parenthesis right parenthesis$ , or $left parenthesis 0 comma 46.8 right parenthesis$ . Therefore, the value of $a$ is approximately $46.8$ . The slope of a line that passes through points $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ can be calculated as $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Since the line of best fit shown for data set E passes approximately through the point $left parenthesis 12 comma 30 right parenthesis$ , it follows that a line of best fit for data set F passes approximately through the point $left parenthesis 12 comma 30 left parenthesis 3.9 right parenthesis right parenthesis$ , or $left parenthesis 12 comma 117 right parenthesis$ . Substituting $left parenthesis 0 comma 46.8 right parenthesis$ and $left parenthesis 12 comma 117 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ yields $\frac{117 - 46.8}{12 - 0}$ , which is equivalent to $StartFraction 70.2 Over 12 EndFraction$ , or $5.85$ . Therefore, the value of $b$ is approximately $5.85$ , or approximately $5.9$ . Thus, $y = 46.8 + 5.9 x$ could be an equation of a line of best fit for data set F.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This could be an equation of a line of best fit for data set E, not data set F.