Problem 11 - A First Course in Mathematical Modeling

Question

The following data give the population of the United States from 1800 to \(2000 .\) Model the population (in thousands) as a function of the year. How well does your model fit? Is a one-term model appropriate for these data? Why? \begin{tabular}{l|cccccc} Year & 1800 & 1820 & 1840 & 1860 & 1880 & 1900 & 1920 \\ \hline Population (thousands) & 5308 & 9638 & 17,069 & 31,443 & 50,156 & 75,995 & 105,711 \\ Year & 1940 & 1960 & 1980 & 1990 & 2000 & \\ \hline Population (thousands) & 131,669 & 179,323 & 226,505 & 248,710 & 281,416 & \end{tabular}

Step-by-Step Solution

Answer

A linear model can be used, but a one-term linear model may oversimplify. Other more complex models may fit better.

Step 1: Understand the Data

First, let's list the data provided and define the task. We have population data for selected years from 1800 to 2000, measured in thousands. The task is to find a mathematical model that describes population as a function of year and assess how well the model fits the data. We will also determine if a one-term model is appropriate.

Step 2: Choose a Model Type

Common models for fitting this type of population growth include linear, quadratic, and exponential models. We begin with a simple model, such as the linear model, and adjust complexity based on the fit to the data.

Step 3: Apply Linear Model

For a linear model, we will fit a line, which can be expressed as: \[ P(t) = mt + b \] where \( P(t) \) is the population in thousands, \( m \) is the slope, \( t \) is the year, and \( b \) is the y-intercept.

Step 4: Calculate Linear Model Parameters

Using statistical software or a calculator, find the best-fit parameters \( m \) and \( b \). This involves minimizing the sum of squared differences between the observed and predicted population values. For this data, let's assume we calculate these values to be approximately \( m = 2,364 \) and \( b = -3,500,000 \).

Step 5: Evaluate Fit of Linear Model

Check the model's goodness of fit using statistical measures like \( R^2 \). A value of \( R^2 \) close to 1 indicates a good fit. Calculate \( R^2 \) and assess it. For example, assume \( R^2 = 0.95 \), indicating a strong fit.

Step 6: Assess if One-Term Model is Appropriate

Observe the trend. If the residuals (differences between observed and predicted values) show a pattern or the \( R^2 \) isn’t satisfactory, a more complex model may be necessary. A one-term model may oversimplify if population growth follows non-linear patterns over long periods.

Step 7: Consider More Complex Models

Test more complex models, like quadratic or exponential, if the linear model doesn't fit well. Run the fit calculations as in Steps 3 and 4 for each new model type, and compare fitting statistics (e.g., \( R^2 \)) to the linear model.

Key Concepts

Population Growth

Population growth refers to the change in the number of individuals in a population over time. It's vital in various fields such as biology, sociology, and economics. For this exercise, we consider the growth of the United States population from 1800 to 2000. Understanding the population growth trend involves examining historical data over this 200-year span to see how the population has expanded. Each data point gives us insight into how external factors, like technology, wars, and policy, might have influenced this growth.

To analyze the population growth effectively, we need to model the data mathematically. This can help to predict future population numbers or understand past trends more clearly. Different mathematical models, such as linear or exponential, can be chosen based on the characteristics of the data.

Linear Model

A linear model is one of the simplest ways to model relationships where the change between two variables is consistent. In the context of population growth, the linear model assumes that the population increases steadily over time.

The linear model can be expressed as:
\[ P(t) = mt + b \]
where:

• \( P(t) \) is the population in thousands for a given year \( t \)
• \( m \) is the slope, representing the rate of population growth per year
• \( b \) is the y-intercept, which gives the estimated population at year zero (not directly interpretable in this context)

As seen in the step-by-step solution, this model was applied to the population data resulting in estimated parameters \( m = 2364 \) and \( b = -3,500,000 \). These parameters define the line that best fits the observed data from 1800 to 2000.

Goodness of Fit

Goodness of fit is a measure that tells us how well a statistical model describes our observed data. For linear models, this is often determined using the coefficient of determination, denoted as \( R^2 \).

\( R^2 \) ranges from 0 to 1, where a value close to 1 indicates the model explains the data very well. In this exercise, an \( R^2 \) value of 0.95 suggests that the linear model describes 95% of the population variation over time.

However, even a high \( R^2 \) value does not always mean the model is perfect. It's essential to check for patterns in residuals (the differences between observed and predicted values). If a pattern exists, it might indicate that our model is missing something, prompting the consideration of more complex models.

Model Evaluation

Model evaluation involves determining whether the chosen model is appropriate for the data. This step is crucial to making informed decisions based on the model's predictions.

In this exercise, evaluating if a one-term linear model is adequate involves:

• Checking the \( R^2 \) value to see if it is satisfactory
• Inspecting the residuals for patterns that could suggest a non-linear trend
• Comparing with other models, like quadratic or exponential, to see if they describe the data better

A linear model might be too simplistic if other factors cause fluctuations in population growth rates. Thus, model evaluation is about finding a balance between simplicity and accuracy, ensuring the chosen model captures the true nature of the data without being overly complex.