As you saw in the world record times data example, linear (line) fits are not always the best way to fit data sets. This lesson will explore how to find exponential fits to various data sets.
We will save the world record times analysis as an exercise for later. Instead, the first data set you will analyze is that of the population of the U.S..
Population growth is a major concern for many countries around the world. These countries fear that one day their population will become so large that they will not be able to provide the basic necessities, such as food and shelter, to their people. Some countries such as China have implemented policies that limit the amount of children a family can have. Other countries in the world, however, have no such policies and their populations continue to spiral.
These are a few of the reasons that it is important to study population growths. In this lesson, we will explore a method of modeling a population by an exponential equation.
This is a data set of the population of the United States since 1805, listed every 10 years. If we could find a model to represent this data then we could predict the population of the U.S. at any time we like. Here is a plot of the U.S. population from 1815-1975. We will use every tenth year starting form 1815 (so 1815, 1825, 1835, etc..) as our data. Also, we will designate 1815 as year 0, so 1825 will be year 1, 1835 as year 2, etc... We do this to make the numbers a little easier to work with in our analysis. So, our data will be years from 0 to 16 and the corresponding population of the United States.
On the plot, the years are plotted on the x-axis and the population on the y.
In this lesson we will use an exponential function to fit two related data sets. To do this, we will use statistics to find the exponential curve that best fits the data.
Each of the following sections should be done in the order presented. They will each have example problems that should be worked by the student. It is important that each section is understood since they build upon each other. This will all lead to a data analysis of a problem that the student will perform on their own. If you need some refreshing on exponentials and logarithms, it is highly recommended to view the REVIEW section and do the exercises.
To do the lesson, download this U.S. Population Excel Spreadsheet. You will need it for the following sections.