CSC: Modeling the USA Population

Chapter 2: 1790-1990

Abstract : In this chapter, we study the relative growth rate of the USA population using as an estimate of the growth rate the average of the right-sided and left-sided average growth rates. We will see that the data break down into three periods for modeling purposes: 1790-1850, 1860-1940, and 1950-1990. For each period, we will fit a function to the data and compute the resulting sum of squared errors as a measure of the goodness of fit.

Prerequisite Knowledge : A conceptual introduction to derivatives; derivatives of polynomial functions; derivative of .

The ultimate goal of this Case Study in Calculus is to model the size of the USA population from census data, and to make predictions about the size in the future. From Chapter 1, we saw that an exponential function gave a good fit for the period 1790-1850. A question we want to answer in this chapter is: Will an exponential function provide a good model for the full range of census data, from 1790 to 1990? To get a start on answering this question, we will use new information we now have, from our study of calculus, about exponential functions and rates of change.

If , where and are constants, then , so (You should supply the missing details when you write up your Conclusions and Summary.) Thus, if we let be the size of the USA population in year , and we plot vs. , then we should expect to see a horizontal line if is indeed an exponential function. We call the relative growth rate of the population and often state it as a percent.

This is all well and good, but our census data is discrete and not continuous. Therefore, we must use an approximation for at each data point. We can use the difference quotients . For example, the right-sided estimate of at 1950 will be (population at 1960 - population at 1950)/10 and the left-sided estimate will be (population at 1940 - population at 1950)/(-10). We will take the average of these two as our estimate of . You should show in your write-up why this equals (population at 1960 - population at 1940)/20. Now, we will use these estimates to generate a table of relative growth rate data which we will also plot.

> dataCensus:=readdata(USAPopCensus17901990,[integer,float]):
ndc:=nops(dataCensus):

relGrRate:=[seq([dataCensus[i][1],100*(dataCensus[i+1][2]-dataCensus[i-1][2])/(20*dataCensus[i][2])],i=2..ndc-1)]:

headers:=["year","relative growth rate (%)"]:
printtable(relGrRate,"Relative Growth Rate (two-sided) Estimates",headers);

plot(relGrRate, style=point);

                       Relative Growth Rate (two-sided) Estimates

                               year       relative growth rate (%)

                               -----------------------------------

                               1800               3.110251

                               1810               2.990725

                               1820               2.951237

                               1830               2.907527

                               1840               3.025701

                               1850               3.093805

                               1860               2.640815

                               1870               2.349204

                               1880               2.303032

                               1890               2.048338

                               1900               1.928601

                               1910               1.643111

                               1920               1.440434

                               1930               1.042477

                               1940               1.082598

                               1950               1.600334

                               1960               1.472151

                               1970               1.148263

                               1980                .989035

From the graph, we see that for modeling purposes the data can be grouped into three parts: 1800-1850; 1860-1940; 1950-1980. Only in the first part does it look like a horizontal line can come close to fitting the data. In the other parts, it still looks like a line, albeit a slanted one, might work. We will pursue this approach and see where it leads.

Fitting the 1800-1850 Data with a Line

Fitting the 1860-1940 Data with a Line

Fitting the 1950-1980 Data with a Line