The sterplot shows the relationship between two variables, x and y . The sterplot has 5 data points. The data

Question

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

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

Accepted Answer

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 ( 0 , b ) is the y -intercept. The graph of an appropriate linear model for this relationship passes near the points ( 1 , 3 ) and ( 9 , 10 ) in the xy -plane. Two points on a line, x 1 , y 1 and x 2 , y 2 , can be used to find the slope of the line using the slope formula, m = y 2 - y 1 x 2 - x 1 . Substituting the points ( 1 , 3 ) and ( 9 , 10 ) for x 1 , y 1 and x 2 , y 2 , respectively, in the slope formula yields m = 10 - 3 9 - 1 , or m = 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 ( 1 , 3 ) , 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 = 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 ( 0 , b ) is the y -intercept. The sterplot 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.

Question 69235d