Use a graphing calculator and perform a regression that fits the data. Consider x to be the number of years after 1900.
About 412.3098 ppm. See solution.
Practice makes perfect
Let x be the number of years after 1900 and y be the carbon dioxide in the Earth's atmosphere (in ppm). From the given table, we can now write a new table with our variables.
x
y
68
324.14
83
343.91
98
367.68
103
376.68
108
385.60
Next, we plot all these points using a graphing calculator. To do so, we push STAT, choose Edit, and enter these values.
Once the values have been entered, we can plot them by pushing 2nd and Y= and choosing one of the plots in the list. Make sure you turn the plot ON, choose scatterplot as the type, and use L1 and L2 as XList and YList. Finally, you can pick whatever mark you want.
By pushing GRAPH the calculator will plot the data set. A standard viewing window might not show the data points, so we can change it if necessary.
Now, let's perform a cubic and quartic regression to approximate the data.
Performing Regressions
By pressing STAT we can find the cubic regression under the CALC menu. If we choose CubicReg the calculator performs a cubic regression test. One line below in the list, we can find QuartReg, which performs a quartic regression.
To determine which model is best, we will plot the results we got and examine how they fit the data.
Making the Prediction
To predict the amount of carbon dioxide in the Earth's atmosphere in 2022 we will use the two regressions we found above. Since x represents the number of years after 1900, we need to substitute x=122.
Let's summarize our predictions in the following table.
Model
Prediction for 2022
Cubic
412.3098 ppm
Quartic
405.5877 ppm
How Confident?
Notice the end behavior of the quartic model is down-and-down. This implies that the amount of carbon dioxide in the Earth's atmosphere will start to decrease at some point, which does not seem reliable (however, let's cross our fingers for this to be true).
In contrast, the cubic model has down-and-up behavior, which implies it continually increases, which sadly seems more reliable. Despite the r^2=1 obtained in the quartic model, the cubic seems to be more likely to represent the carbon dioxide amounts over time.
