Abstract

This report analyses the exponential growth of the US population from 1790 to 1850 using statistical programming technique where we will be applying a non-linear least squares regression model to an exponential growth equation to determine the exponential growth constant. This analysis provides a significant first-order exponential growth pattern with the popular increasingly over time. The exponential growth model’s parameters were determined, assessed and evaluated and such model can be used to determine data in many different fields like finance or to track the spread of a disease.

Introduction

The data set we were provided was the population growth of the US from 1790 to 1850, which was the perfect data set for an exponential growth as it was a rapidly growing data set due to favorable conditions but with minimal limiting factor, giving us the steady, accelerating increase which is a nice fit with an assumptions of an exponential growth model.

Graphing the exponential curve

## Given is the data for the population growth of the US from 1790 to 1850 where the year 1790 is set to 0 and accordingly 

Year <- c(0,10,20,30,40,50,60)
Year
## [1]  0 10 20 30 40 50 60
##Given is the population in millions 

Pop <- c(3.929, 5.308, 7.240, 9.638, 12.866, 17.069, 23.192)
Pop
## [1]  3.929  5.308  7.240  9.638 12.866 17.069 23.192
##Graphing the population growth of the US from 1790 to 1850
plot(Year, Pop, xlab = 'Years (from 1790 to 1850)',ylab='Population (in millions)',main='Population Growth of the US from 1790-1850',col="blue")

Exponential model fitting

We will be using NLS2 function to estimate the parameters of data by using the equation of exponential growth to the data. The equation is as given below. \[A = A0\times exp^(k*year)\] where \(t\) is the time in years \(A\) is the population at time t \(A0\) is the initial population at 1790 (denoted as 0) \(k\) is the growth constant \(exp^(k*year)\) is the exponential growth function as a growth function

For non-linear squares regression, the package ‘nls2’ from R markdown was used to find a curve that has the best fit to the given data points.

##nls2 package

library(nls2)
## Loading required package: proto
fitline <- nls2(Pop~A0*exp(k*Year), start = list(A0=3.97, k=0.05),)

#Parameters
summary(fitline)
## 
## Formula: Pop ~ A0 * exp(k * Year)
## 
## Parameters:
##     Estimate Std. Error t value Pr(>|t|)    
## A0 3.9744064  0.0407277   97.58 2.14e-09 ***
## k  0.0293421  0.0002023  145.02 2.96e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.09817 on 5 degrees of freedom
## 
## Number of iterations to convergence: 5 
## Achieved convergence tolerance: 5.311e-08
plot(Year, Pop, xlab = 'Years (from 1790 to 1850)',ylab='Population (in millions)',main='Population Growth of the US from 1790-1850',col="blue")

#Fitted exponential curve line 
lines (Year,predict(fitline),col="pink")

The result of the manual calculation of k is 0.030 per year and A0 is 3.929 million.

Conclusions

The precise parameters we obtained from fitting the exponential data set was 0.029 for k which was not too far off from our manual calculation of 0.030. The initial value was calculated to be 3.929 millions and the obtained results through R was 3.9744, closely related. This result demonstrate the exponential model that perfectly describes the population growth of the US from 1790 to 1850.