Question

For the following exercises, use the population of New York City from 1790 to 1860 , given in the following table. $$ \begin{array}{|l|l|} \hline \text { Years since } 1790 & \text { Population } \\ \hline 0 & 33,131 \\ \hline 10 & 60,515 \\ \hline 20 & 96,373 \\ \hline 30 & 123,706 \\ \hline 40 & 202,300 \\ \hline 50 & 312,710 \\ \hline 60 & 515,547 \\ \hline 70 & 813,669 \\ \hline \end{array} $$ Using a computer program or a calculator, fit a growth curve to the data of the form \(p=a b^{t}\).

Short answer

Fit an exponential model using regression to find \( a \) and \( b \) from the data.

Step-by-step solution

Understand the Exponential Model

The formula given is \( p = a b^{t} \), where \( p \) is the population at time \( t \), \( a \) is the initial population when \( t = 0 \), and \( b \) is the growth factor. The goal is to find the values of \( a \) and \( b \) that best fit the data.

Set up the Data Points

From the table, obtain the pairs \((t, p)\): (0, 33131), (10, 60515), (20, 96373), (30, 123706), (40, 202300), (50, 312710), (60, 515547), and (70, 813669).

Logarithmic Transformation

To linearize the exponential function, take the logarithm of both sides: \( \log(p) = \log(a) + t\log(b) \). This is now in the form of a linear equation \( y = mx + c \), where \( y = \log(p) \), \( m = \log(b) \), \( x = t \), and \( c = \log(a) \).

Calculate \( \log(p) \) for Each Data Point

Calculate \( \log(p) \) for each population value, enabling the transformation of the original exponential model into a linear model. Use the base 10 logarithm for consistency.

Input Data for Linear Regression

Use a calculator or computer software capable of performing linear regression analysis, inputting the \( t \) values and the calculated \( \log(p) \) values to determine the best-fit line.

Determine Linear Regression Parameters

The regression analysis will produce parameters \( c \) (the y-intercept) and \( m \) (the slope), where \( c = \log(a) \) and \( m = \log(b) \).

Extract \( a \) and \( b \) from Regression Output

Exponentiate to solve for \( a \) and \( b \): \( a = 10^c \) and \( b = 10^m \). These give the parameters for the exponential model.

Verify the Model

Use \( a \) and \( b \) to calculate predicted populations \( p \) for each \( t \) and compare these with actual data to ensure a good fit.

Key concepts

Population Growth

Population growth is a fascinating area of study in mathematics and science. It refers to how populations of living organisms, like humans or animals, increase over time. When a population grows at a constant rate per unit time, we call it exponential growth. In other words, the larger the population gets, the faster it grows. For instance, the population growth of New York City from 1790 to 1860 showed an exponential trend as it moved from 33,131 people to 813,669 in 70 years.
This growth can be due to various factors such as increased birth rates, decreased death rates, or migration. Exponential growth models, like the one used for New York City's population, often serve as simplified representations of real-world population dynamics.
Observing historical population data allows us to predict future trends, which is crucial for resource planning, infrastructure development, and policy-making.

Linear Regression

Linear regression is a statistical method used to find the best-fitting line through a set of data points. This method helps in understanding relationships between variables, by finding a line that minimizes the difference between the data points and the line itself. In the context of population growth modeling, linear regression helps to simplify complex exponential relationships.
After transforming the exponential data, such as population numbers over time, into a linear form through logarithmic transformation, linear regression can identify the slope and intercept of the best-fit line. These parameters are crucial in defining mathematical relationships, which we can use to make predictions.

• Slope (\(m\)): Represents the change in the log of the population per unit time.
• Intercept (\(c\)): The log of the initial population size when time is zero.

Linear regression not only aids in understanding trends but also helps in predicting future values.

Logarithmic Transformation

To better fit an exponential growth model to data, we often need a logarithmic transformation. This involves taking the logarithm of each data point in an exponential dataset, converting it into a linear form. The transformation equation starts with the exponential model: \( p = a b^t \).
When we take the logarithm on both sides, it becomes \( \log(p) = \log(a) + t \log(b) \), which is a linear equation of the form \( y = mx + c \) in terms of the log of the population \( \log(p) \) and time \( t \).
This transformation is essential because linear equations are easier to handle, interpret, and fit a line to through linear regression. Performing this transformation allows the exponential model's parameters \( a \) and \( b \) to be extracted from linear regression analysis, which then can be used for making accurate predictions.

Data Fitting

Data fitting involves the process of creating a model that best represents the relationship within a dataset. When studying population growth, data fitting ensures that our mathematical model projects the population changes accurately. Techniques like non-linear and linear regression are often used to fit data, after transforming it appropriately.
When applying the exponential growth model to historical data such as New York's population, the goal of data fitting is to find the parameters that make the model as close as possible to the observed data points. This involves transforming the data when necessary, using models and transformation techniques like logarithmic transformation, and then reevaluating the model's accuracy.
This process is crucial, as a well-fitted model allows analysts to extrapolate future values or understand past behaviors based on changing conditions. Thus, data fitting not only serves numerical entirety but also provides insights for crucial decision-making in planning and analysis.