Question

Population The numbers \(P\) (in millions) of people age 18 and over in the United States for the years 1996 to 2005 are shown in the table. (Source: U.S. Census Bureau) $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population, } P \\ \hline 1996 & 199.2 \\ \hline 1997 & 201.7 \\ \hline 1998 & 204.4 \\ \hline 1999 & 207.1 \\ \hline 2000 & 209.1 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population, } P \\ \hline 2001 & 212.5 \\ \hline 2002 & 215.1 \\ \hline 2003 & 217.8 \\ \hline 2004 & 220.4 \\ \hline 2005 & 222.9 \\ \hline \end{array} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t=6\) correspond to \(1996 .\) (b) Use the regression feature of a graphing utility to find a linear model, a quadratic model, and a cubic model for the data. (c) Use a graphing utility to graph each model separately with the data in the same viewing window. How well does each model fit the data? (d) Use each model to predict the year in which the population is about \(231,000,000 .\) Explain any differences in the predictions.

Short answer

This problem involved an analysis of population data using scatter plotting and different regression models. The results would shed insight on predicting future population considering linear, quadratic, and cubic models. The exact numerical answer and the predicted year may vary depending on the calculated regression models.

Step-by-step solution

Create a Scatter Plot

First, to create a scatter plot the given data should be plotted on a coordinate grid with \(t=6\) corresponding to 1996 as the x-coordinate and the population of that year as the y-coordinate. This will help in visualizing the data.

Find the Models

Then, the regression feature of the graphing utility should be used to find a linear model, a quadratic model, and a cubic model for the data. This can be typically done by entering all data points and selecting the type of regression model wanted.

Compare the Model Fits

Having found the models, the task now is to graph each model separately with the data in the same viewing window to compare how well each model fits the data. This can be evaluated by how closely they follow the data points in scatter plot.

Predict the Future Population

Finally, use the models found in step 2 to predict the year in which the population is about 231,000,000. Analyze the differences in the predictions from the three models and explain why they might occur.

Key concepts

Scatter Plot

When tackling problems relating to population growth, a terrific starting point is often a scatter plot. Visualizing the data by placing each year's population figure on a graph helps us discern patterns and anomalies within the population changes over time.

To create a scatter plot, you would use a set of axes where the x-axis typically represents time, and the y-axis represents the population. Each data point on the graph corresponds to a specific year's population statistic. For the data given from 1996 to 2005, where each increment on the x-axis could represent a year, and the corresponding population on the y-axis is measured in millions. By seeing the data points spread out across the plot, you can begin to understand the dynamics of the population growth and prepare for the next step—choosing a suitable model to describe that growth.

Linear Regression

Linear regression is a statistical method that you can use to create a model predicting population growth. In simple terms, it tries to draw a straight line that best fits all the data points on the scatter plot.

This straight line is an equation of the form \( P = mt + b \), where \( P \) is the population, \( t \) is the time (such as the year), \( m \) is the slope of the line, indicating the rate of population growth per year, and \( b \) is the y-intercept, representing the estimated population at the starting point of the data. A linear regression model assumes a constant rate of growth—whether the population consistently increases or decreases over time. It’s best-suited for data that doesn’t deviate much from a straight path.

Quadratic Model

To capture more complexity in population growth, such as acceleration or deceleration in the rate of growth, a quadratic model is often more suitable. This model is reflected in a parabolic curve on the scatter plot, represented by the equation \( P = at^2 + bt + c \).

The term \( at^2 \) introduces a squared relationship between time and population, capturing the effect of growth rate changes over time. In certain instances where linear models fail to account for this acceleration or deceleration, a quadratic model provides a more accurate representation of the population trends. Hence, the quadratic model can provide a significant advantage when making medium-range predictions.

Cubic Model

For even more detailed modeling, when the population growth rate shows multiple fluctuations, a cubic model might be the best fit. This model follows a cubic equation: \( P = at^3 + bt^2 + ct + d \).

With the cubic term \( at^3 \), the model can account for various growth patterns, including changes in the direction of acceleration or deceleration. It can identify inflection points where the rate of growth not only speeds up or slows down but also changes from one to the other. The complexity of the cubic model allows for a closer fit to real-world data that exhibits multiple trends over time but requires careful handling to avoid overfitting—where the model becomes too tailored to the sample data and less predictive for future trends.

Predictive Analysis

Last comes the predictive analysis—using the models derived from the scatter plot to forecast future population sizes. After establishing linear, quadratic, and cubic models, the goal is to estimate the time it will take for the population to reach a certain level, in this case—the year the population will reach approximately 231 million people.

Predictive analysis involves plugging the population value into the equation of each model and solving for \( t \) (the year). Each model will yield a different prediction. These differences arise because each model interprets the trend in the historical data differently. The linear model assumes a constant growth rate, the quadratic considers a changing rate, and the cubic accounts for multiple changes in the trend. When interpreting the results, it's crucial to consider which model is most suitable based on the nature of the population growth you've observed, and the timeframe for your prediction.