Question

Predicted concentrations of atmospheric carbon dioxide (CO2) in parts per million (ppm) are shown in the table below. (These concentrations assume that current trends continue.) 2000 2050 2100 2150 2200 CO (ppm) 364 467 600 769 987 Year a) Use the graphing calculator to make a scatterplot of the data. Let x represent years after 2000. Does the data follow a linear trend? Explain. b) The graphing calculator allows you to obtain different regression models for the given data (Stat> Calc) and their corresponding coefficients of determination r2. The closest r2 is to 1, the better your model is. Determine the following regression models and r2, making sure you round your coefficients to four decimal places) c) QuadReg y Quadratic Regressiorn ExpReg y Exponential Regression d) Graph each model with the data . Can you determine, by looking at the graphs, which model fits the data best? What can you use to help you choose the best model? e) The exponential model that fits the data is Use the previous model to predict CO2 concentrations for the year 2025 f)

Answer 1

a) x will be taking values (0,50,100,150,200) and the concentration (c) will be taking values (364, 467, 600, 769, 987). Using these values we obtain the plot:

6007008009001000 400 500

b) If we fit a linear regression to the data then it can be found that the equation to it will be: y= 327.8 + 3.096*x.

And when we draw this line along with the scatter plot then,

From the graph it is evident that, most of the values will be over estimated if we continue with linear regression. Hence, we can say that the data does not follow a linear trend.

c) Quadratic Regression:  

y= 365.8000 + 1.5760 x + 0.0076 x^2

r^2 = 0.9998496

Exponential Regression:

y= 364.0445*(1.005)^x

r^2 = 0.9999983

d)

In the above graph, the red line denotes the exponential and the black one denotes the quadratic regression lines. Since the two regression lines are almost overlapping, it is hard to say which one is a better fit.

However we can use the value of the co-efficient of determination to make the decision. Since the co-efficient of determination is slightly greater for the exponential regression, we say that the exponential regression is slightly better.

e) To predict the concentration for the year 2025, we have to put x= 2025-2000=25 in the equation: y= 364.0445*(1.005)^x

Hence, y= 364.0445*(1.005)^(25)= 412.388