Question

Population The populations \(P\) of the United States (in thousands) from 1990 to 2005 are shown in the table. (Source: U.S. Census Bureau)$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population } \\ \hline 1990 & 250,132 \\ \hline 1991 & 253,493 \\ \hline 1992 & 256,894 \\ \hline 1993 & 260,255 \\ \hline 1994 & 263,436 \\ \hline 1995 & 266,557 \\ \hline 1996 & 269,667 \\ \hline 1997 & 272,912 \\ \hline \end{array} $$$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population } \\ \hline 1998 & 276,115 \\ \hline 1999 & 279,295 \\ \hline 2000 & 282,403 \\ \hline 2001 & 285,335 \\ \hline 2002 & 288,216 \\ \hline 2003 & 291,089 \\ \hline 2004 & 293,908 \\ \hline 2005 & 296,639 \\ \hline \end{array} $$(a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=0\) corresponding to 1990 . (b) Use the regression feature of a graphing utility to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln b}\) to rewrite the model as an exponential model in base \(e\). (c) Use the regression feature of a graphing utility to find a linear model and a quadratic model for the data. (d) Use a graphing utility to graph the exponential model in base \(e\) and the models in part (c) with the scatter plot. (e) Use each model to predict the populations in 2008 , 2009 , and 2010 . Do all models give reasonable predictions? Explain.

Short answer

The short answer here is not definitive, as it requires the graphing utility to do the actual calculations and predictions, and the 'reasonability' of the model outputs is a somewhat subjective judgment. However, in general, the model which appears to follow the existing data trend most closely is likely to yield the most reasonable predictions for future years.

Step-by-step solution

Creation of Scatter Plot

First, enter the given data into the graphing utility, with the year in one column and the corresponding population in another. Set \(t = 0\) for the year 1990, and increment it by one for each following year. Use the scatter plot feature to graph the data.

Exponential Regression

Once the scatter plot is generated, apply the regression feature of the graphing utility to fit an exponential model. This will render an equation in the form \(P=ab^t\). Use the Inverse Property \(b = e^{\ln b}\) to rewrite it as an exponential model in the base \(e\).

Linear and Quadratic Regression

Similarly, apply the linear regression and quadratic regression features of your graphing utility to come up with a linear model \(P=mt+c\) and a quadratic model \(P=at^2+bt+c\) respectively.

Graphing the Models

Next, graph the exponential, linear and quadratic models simultaneously on top of the scatter plot, using your graphing utility.

Making Predictions

Using each of the three models, predict the population for the years 2008, 2009 and 2010. This again, can be done using your graphing utility.

Model Comparison

Compare the predicted values from each model. Not all models may yield reasonable predictions - it is up to you to decide which model's predictions seems to fit best with the actual data trend.

Key concepts

Scatter Plot

When studying population data, a scatter plot is a valuable tool in visualizing the relationship between two variables. It's a type of graph that shows individual data points representing two related quantities—the year and the population in this case. Each point represents the population at a specific year, with the x-axis often representing time and the y-axis representing the quantity of interest, such as the population.

Creating a scatter plot is straightforward. For every year, mark a point on the graph where the x-coordinate corresponds to the time since the start year (with 1990 as the baseline, or t=0), and the y-coordinate corresponds to the population in thousands. The resulting pattern of points can give an immediate visual indication of the data trends, such as increases, decreases, or cyclical patterns. Scatter plots are the first step in determining the type of regression model to apply for predictions.

Exponential Regression

Moving to more advanced analysis, exponential regression is used when data increases or decreases at a constantly changing rate. It is especially useful for modeling population growth, radioactive decay, and many other natural phenomena. The general form of an exponential equation is
\( P = ab^t \).

Using the scatter plot data, graphing utilities can calculate the best-fitting exponential curve. This involves finding the parameters 'a' and 'b' that minimize the differences between the predicted and actual values. In algebra, the concept of exponential growth is often expressed using the natural base 'e'. By using the Inverse Property, the model can be rewritten as
\( P = ae^{(t \times \text{ln}(b))} \).

This version of the equation allows for more straightforward computations, especially when using e-based logarithmic transformations to solve or manipulate the model.

Linear and Quadratic Regression

In contrast to the multiplicative nature of exponential growth, linear and quadratic regressions are additive models. When data points form a pattern that closely resembles a straight line, a linear model is applicable, expressed as
\( P = mt + c \),
where 'm' represents the slope of the line, and 'c' is the y-intercept, or the population value when t=0. This model implies a constant rate of change over time—a steady increase or decrease in population with each passing year.

For data that changes in rate over time—not consistent enough for a straight line—a quadratic model might be more appropriate. This model is represented by the equation
\( P = at^2 + bt + c \),
with 'a', 'b', and 'c' being the coefficients that define the parabola's curvature, slope, and position, respectively. A quadratic model can capture patterns such as acceleration and deceleration in population growth.

By fitting these models to the data with graphing utilities, we can graph the predicted values and compare them to the exponential model and the actual scatter plot to determine which model best represents the historical data and can predict future trends most accurately.