The U.S. Population

In this demonstration we will try to determine what type of equation best describes population growth. The two types of functions that we have studied so far are lines and quadratics. Is population growth modeled by either of these types of functions? To answer this question, we are going to compare what happens when we use a line and a quadratic function to model the population growth in the United States. To make the numbers more manageable, let x=0 for the first year in the table in the table below (1790). You need to rewrite the other years in terms of how many years they are after 1790. For example, the year 1800 will be entered as x=10.

Part 1: Use a statistical package or a graphing calculator to make a scatterplot with the number of years since 1790 as the independent variable and with the population as the dependent variable. Fit a line to this data. What is the R² value for the line? What is the pattern of the slopes between each consecutive pairs of points? How does this tell you that the data is not linear?

Part 2: Use a statistical package or a graphing calculator to fit a quadratic function to this data set. What is the R² value for the quadratic? What is the adjusted R² value for the quadratic? (It may not be possible to answer this question if you are using a graphing calculator.) Use this equation to predict the United States population in 1700 and in the next year. Does your prediction for 1700 have any basis in reality? Why does the quadratic not predict the population in 1700 with any degree of accuracy?

As you can see from Part 2, a parabola does not always accurately describe population growth. An equation that does a better job modeling population growth, along with describing many other situations, is an exponential equation.