The given bar graph shows consumption, in millions of tons, of several fabric materials in 2010 and 2019 in a factory in the United States. In a scatterplot of this data, where the material consumption in 2010 is plotted on the 𝑥-axis and the material consumption in 2019 is plotted on the 𝑦-axis for each of the different materials. How many data points would be below the line 𝑦 equals 𝑥?
Let’s think about a scatterplot of this data. We have our 𝑥-axis and our 𝑦-axis. And then let’s add the line 𝑦 equals 𝑥 to this graph. We could mark the point one, one; two, two; and three, three. Because whatever 𝑥 is, 𝑦 is the same thing. And then we sketch our line.
We’re interested in considering how many data points would fall below the line of 𝑦 equals 𝑥. That means we’ll want to know how many data points would fall in this space. The data points below 𝑦 equals 𝑥 are all the places where, in the coordinates, the 𝑥-value is greater than the 𝑦-value. If the 2010 fabric consumption is the 𝑥-axis and the 2019 consumption is on the 𝑦-axis. The data points that would fall below the line 𝑦 equals 𝑥 is the places where 2010 consumption is larger than 2019 consumption.
Since we see that the 2010 is the darker grey and 2019 is the lighter grey. We want to consider the fabrics where the lighter grey is less than the darker grey. If we consider the jeans consumption, it increased from 2010 to 2019. The khaki consumption decreased from 2010 to 2019. Since khaki is greater in 2010 than 2019, it falls below the line 𝑦 equals 𝑥. There’s an increase in velvet use and an increase in polyester. But the cotton use did decrease from 2010 to 2019. Because cotton consumption in 2010 was greater than cotton consumption in 2019, it would also fall below the line 𝑦 equals 𝑥. And we’ve shown that two data points fall below this line.
Using this method, we haven’t actually created a scatterplot. We’ve only considered where each of these five materials would have fallen if we did make a scatterplot. If you wanted to check if our first method was correct, we could make a full scatterplot of the data by labeling our 𝑥- and 𝑦-axis and then plotting the five points. Jeans had a usage of two million tons in 2010 and between 2.5 and three million tons in 2019. Khaki had a little bit more than one-half a million tons in 2010 and a little less than half a million tons in 2019. We will continue with graphing the other three points. Once you have all five of these points, you need to draw the line 𝑦 equals 𝑥. And again, we see that cotton and khaki are below the line 𝑦 equals 𝑥. And it confirms our original conclusion.
Two data points would fall below the line 𝑦 equals 𝑥, given these conditions.