US Population Growth (1790-1850)

A non-linear least squares fit was used to determine the US population growth constant (k) from 1790 to 1850. First, a plot of years versus population (in millions) was created to visualize the growth trend. For ease of readability and data processing, the years on the x-axis are represented as the number of years since 1790.

#Defining Variables
year <- c(0, 10, 20, 30, 40, 50, 60) #x
pop <- c(3.929, 5.308, 7.240, 9.638, 12.866, 17.069, 23.192) #y

#Plotting x vs. y
plot(year,pop,
     xlab='Years (Starting from 1790)',
     ylab='Population in Millions',
     main='US Population Growth (1790-1850)')

To model this growth trend, an exponential growth model was applied:

\[ \text{Population} = \text{popi} \cdot e^{k \cdot \text{year}} \] popi is the initial US population at year 0 (1790)
k is the growth rate constant
year is the number of years since 1790

A non-linear least squares fit line was added to the exponential model to estimate the growth constant k.

#nls2 fitting
plot(year,pop,
     xlab='Years (Starting from 1790)',
     ylab='Population in Millions',
     main='US Population Growth (1790-1850)')

library(nls2)

## Loading required package: proto

populationmodel <- nls(pop ~ popi*exp(k * year),
              start = c(popi = 3.929, k = 0.01)) 
summary(populationmodel)

## 
## Formula: pop ~ popi * exp(k * year)
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## popi 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: 4 
## Achieved convergence tolerance: 4.981e-08

lines(year,predict(populationmodel),col= 'red')

The nls2 model provides a strong fit to the exponential population model, with minimal noise or outlier data points detected. According to the non-linear least squares summary, the growth rate (k constant) of the US population from 1790 to 1850 is estimated to be 0.0293421.