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

The figure presents a scatterplot titled “Railroad Museum Visitors.” The horizontal axis is labeled “Years since 1968,” and the numbers 0 through 15, in increments of 5, are indicated. The vertical axis is labeled “Annual visitors,” and the numbers 0 through 80,000, in increments of 20,000, are indicated. There are 13 data points indicated on the scatterplot. The data points begin at the point with approximate coordinates 0 comma 5,000, and trend upward and to the right until they end at the point with approximate coordinates 12 comma 75,000. A line of best fit is drawn. The line begins at the point with approximate coordinates 0 comma 16,000, and slants upward and to the right. It passes through the point with approximate coordinates 5 comma 40,000, and continues upward until it ends at the point with approximate coordinates 12 comma 74,000.

The scatterplot above shows the number of visitors to a railroad museum in Pennsylvania each year from 1968 to 1980, where t is the number of years since 1968 and n is the number of visitors. A line of best fit is also shown. Which of the following could be an equation of the line of best fit shown?

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Explanation

Choice A is correct. An equation of a line of best fit can be written in the form $y equals, a, plus b x$ , where a is the y-intercept of the line and b is the slope. In the scatterplot shown, the line of best fit intersects the y-axis just over halfway between 10,000 and 20,000, or approximately 16,000. The line of best fit also intersects the graph at $the point with coordinates 5 comma 40,000$ . Using the slope formula $b equals, the fraction with numerator y sub 2 minus y sub 1, and denominator x sub 2 minus x sub 1, end fraction$ and two points that lie on the graph such as $the point with coordinates 5 comma 40,000$ and $the point with coordinates 0 comma 16,000$ , the slope can be approximated as $the fraction with numerator 40,000 minus 16,000, and denominator 5 minus 0, end fraction$ , or 4,800. Only choice A has a y-intercept near the estimate of 16,000 and a slope near the estimate of 4,800. Therefore, an equation of the line of best fit could be $n equals, 16,090 plus 4,680 t$ .

Choice B is incorrect because the values for the slope and the y-coordinate of the y-intercept are switched. Choice C is incorrect because the value for the slope is approximately double the actual slope. Choice D is incorrect because the values for the slope and the y-intercept are switched and because the slope is approximately double the actual slope.