In this project, you will use an arithmetic growth model to approximate real data published in the form of a graph or chart. Before meeting, each member of the group should individually find a graph in a magazine or newspaper that appears to be approximated by a straight line. Avoid graphs for which one axis refers to some type of categories. For example, a graph showing medicare expenditures by state would not be acceptable, because one of the axes of the graph, the one showing states, is a category. You want to have a graph with numerical axes, so that it will make sense to extend the graph beyond the original data shown. A graph that has one axis for time and the other for some other numerical variable is ideal for this activity.
As a group, pick the graph that seems best suited for an arithmetic growth model. It should be the one that is closest in appearance to a straight line. For the graph that you pick, create a table of x and y values, either using data values given in the graph, or if necessary, estimating as accurately as possible from the data points on the graph. For this data set, formulate an arithmetic growth model. The model must include definitions for the variables, a difference equation, a graph of both the model data and the difference equation, [and a functional equation if you can find one].
You can test out how well different variations on your model fit the data points by plugging in known data values. A useful measure to determine if your model is reasonable is to look at the deviations of the model from the known data (the error terms). The best model is usually the model that has the smallest error for each point, or the smallest error overall (or sometimes we use the criterion that we want no error to be greater than a certain amount). Note that the smallest error for all points may not be the smallest for any particular point. So we might have a model that fits the data perfectly for some data values, but is too large for some of the data values.
Part of the project assignment is to write a report about your model. The report should discuss all of the things mentioned above, including the definitions of your variables, the difference (and functional equations if you found one), and graphs of the original data and the model. You should also discuss how accurate the model is, in terms of the sizes of the errors you observed. In addition, use the model to make projections beyond the original data. For example, if your original graph showed data for several different years, you can use the model to predict what will happen for future years. Be sure to include a discussion of how reliable you think these predictions are.