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

The scatterplot shows the relationship between two variables, $x$ and $y$ .

Which equation is the most appropriate linear model for this relationship?

$y = - 0.9 x - 2.2$

$y = - 0.9 x + 2.2$

$y = - 0.9 x$

$y = 0.9 x + 2.2$

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Explanation

Choice D is correct. A linear model can be written in the form $y = m x + b$ , where $m$ is the slope of the graph of the model in the xy-plane and $left parenthesis 0 comma b right parenthesis$ is the y-intercept. The graph of an appropriate linear model for this relationship passes near the points $left parenthesis 1 comma 3 right parenthesis$ and $left parenthesis 9 comma 10 right parenthesis$ in the xy-plane. Two points on a line, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , can be used to find the slope of the line using the slope formula, $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Substituting the points $left parenthesis 1 comma 3 right parenthesis$ and $left parenthesis 9 comma 10 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the slope formula yields $m = \frac{10 - 3}{9 - 1}$ , or $m equals 0.875$ . Therefore, the value of $m$ for an appropriate linear model is approximately $0.875$ . Substituting $0.875$ for $m$ in $y = m x + b$ yields $y = 0.875 x + b$ . Since an appropriate linear model passes near the point $left parenthesis 1 comma 3 right parenthesis$ , the approximate value of $b$ can be found by substituting $1$ for $x$ and $3$ for $y$ in the equation $y = 0.875 x + b$ , which yields $3 = (0.875) (1) + b$ , or $3 = 0.875 + b$ . Subtracting $0.875$ from both sides of this equation yields $2.125 equals b$ . Therefore, the value of $b$ for an appropriate linear model is approximately $2.125$ . Thus, of the given choices, $y = 0.9 x + 2.2$ is the most appropriate linear model for this relationship.

Alternate approach: A linear model can be written in the form $y = m x + b$ , where $m$ is the slope of the graph of the model in the xy-plane and $left parenthesis 0 comma b right parenthesis$ is the y-intercept. The scatterplot shows that as the x-values of the data points increase, the y-values of the data points increase, which means the graph of an appropriate linear model has a positive slope. Of the given choices, $y = 0.9 x + 2.2$ is the only linear model whose graph has a positive slope.

Choice A is incorrect. The graph of this model has a negative slope, not a positive slope.

Choice B is incorrect. The graph of this model has a negative slope, not a positive slope.

Choice C is incorrect. The graph of this model has a negative slope, not a positive slope.