CSC: Modeling the USA Population

Chapter 1: 1790-1850

[Thanks to Eugene Demidenko, Research Assistant Professor, Community and Family Medicine, Dartmouth Medical School, for suggesting the Case Study and consulting on it.]

Abstract : In this chapter, we will be using elementary functions in a real application with real data. We will be modeling USA population census data for the period 1790-1850 with polynomial and exponential functions, and comparing their goodness of fits.

Prerequisite Knowledge : The elementary functions.

The purpose of this Case Study in Calculus is to model the USA population from census data to make predictions about the size in future years. We begin by reading in the Census data from 1790-1990 and extracting from it the data for the period we are going to study in this chapter, namely, 1790-1850. We will exhibit these latter data both numerically and graphically.

> dataCensus:=readdata(USAPopCensus17901990,[integer,float]):
data1:=[seq(dataCensus[i],i=1..7)]:

with(math3):
intdata1:=[seq([data1[i][1],round(data1[i][2])],i=1..7)]:
headers:=["year","Population"]:
printtable(intdata1,"USA Population (in thousands)", headers);

plot(data1,style=point,title="USA Census Data 1790-1850");

                             USA Population (in thousands)

                                year            Population

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

                                1790                3929

                                1800                5297

                                1810                7224

                                1820                9618

                                1830               12901

                                1840               17120

                                1850               23261

From the graph, it seems reasonable to try to fit either a polynomial or an exponential to these points. We will try each approach and compare them.

Fitting the data with a Polynomial Function

Fitting the Data with an Exponential Function

Which Function Gives a Better Fit?