Question

Cost of Dental Care The amount that \(\$ 100\) worth of dental care at \(1982-1984\) prices would cost in a different year is given by a CPI (Consumer Price Index). The CPIs for dental care in the United States for the years 1996 to 2005 are shown in the table. (Source: U.S. Bureau of Labor Statistics) $$ \begin{array}{|c|c|} \hline \text { Year } & \text { CPI } \\ \hline 1996 & 216.5 \\ \hline 1997 & 226.6 \\ \hline 1998 & 236.2 \\ \hline 1999 & 247.2 \\ \hline 2000 & 258.5 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline \text { Year } & \text { CPI } \\ \hline 2001 & 269.0 \\ \hline 2002 & 281.0 \\ \hline 2003 & 292.5 \\ \hline 2004 & 306.9 \\ \hline 2005 & 324.0 \\ \hline \end{array} $$ (a) Use a spreadsheet software program to create a scatter plot of the data. Let \(t\) represent the year, with \(t=6\) corresponding to \(1996 .\) (b) Use the regression feature of a spreadsheet software program to find a linear model, a quadratic model, a cubic model, and a quartic model for the data. (c) Use each model to predict the year in which the CPI for dental care will be about \(\$ 400\). Then discuss the appropriateness of each model for predicting future values.

Short answer

This exercise does not provide specific numerical answers considering it needs practical application of a spreadsheet software for completion. However, the short answer should include the year predictions for each model (linear, quadratic, cubic, and quartic) and a brief discussion suggesting which model seems most appropriate for predicting the CPI of future years.

Step-by-step solution

Create a Scatter Plot

To begin, open a preferred spreadsheet software, enter the given data into respective columns, one column for years and the other for CPI. Set \( t=6 \) for the year 1996 and increase \( t \) by 1 every year afterwards. A scatter plot can be created based on the given values.

Find Different Models

Next, use the regression feature of your spreadsheet software. Find a linear, quadratic, cubic, and quartic model for this data. This step provides different mathematical models describing the behavior of the data.

Predict CPI

Finally, use each of the four models obtained to predict the year in which the CPI for dental care will be equal to $400. By setting the model equations equal to 400, we can solve for \( t \) to find the respective years for each model.

Discuss the Appropriateness of each Model

With the predictions, evaluate and discuss the appropriateness of each model. The appropriateness can then be determined by considering the feasibility of the predictions and pattern of the original data.

Key concepts

Scatter Plot Analysis

Scatter plot analysis is a graphical representation used to show the relationship between two variables. In the context of predicting CPI for dental care, it's the first step in understanding the trend of how the CPI has changed over the years. By placing the years on the x-axis (with a new variable, t, where t=6 corresponds to the year 1996) and the CPI values on the y-axis, a scatter plot can provide a visual indication of whether there's a linear, quadratic, or more complex relationship.

To enhance comprehension, students may be encouraged to label the axes clearly, use a consistent scale, and perhaps even color-coordinate different years for easier visualization. In educational settings, it's crucial to guide students through the process of creating scatter plots to ensure they can effectively interpret the graphical data presented.

Regression Models

Regression models are mathematical tools used to describe the relationship between variables and predict future values. In our exercise, linear, quadratic, cubic, and quartic regression models are constructed from the data points on the scatter plot using spreadsheet software.

• A linear regression model assumes a straight-line relationship between the two variables.
• A quadratic regression model includes an x² term, allowing for a parabolic curve that can model non-linear data more effectively.
• A cubic regression model includes an x³ term and can capture more complex relationships and inflection points.
• A quartic model adds an x⁴ term and can fit even more intricate patterns.

Selecting the appropriate model depends on the pattern of the data points on the scatter plot. A careful analysis of the residual plots can also greatly assist in deciding which model best fits the data without overfitting.

Predictive Modeling in Economics

Predictive modeling in economics utilizes various statistical techniques to forecast economic events, such as the fluctuations in the CPI for dental care. This exercise employs regression models as predictive tools, assuming that historical data can help estimate future trends.

Economic prediction models must take into account the potential for changing conditions, outliers, and unexpected variables. In our scenario, predicting when the CPI will reach $400 using the different regression models showcases an application of these models in an economic context. For instance, while simple linear prediction is intuitive, it may not always capture the nuances of economic data, highlighting the importance of understanding and selecting the right model for accurate predictions.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of real-world situations to make predictions or gain insights. In the dental care CPI prediction exercise, models ranging from simple to complex (linear to quartic) represent different assumptions about the underlying pattern of the data.

For effective mathematical modeling, it's essential to balance simplicity and accuracy. Simpler models are easier to understand and use, while more complex models may fit the historical data better but can also lead to overfitting. It's crucial for students to learn that the best model is not always the one that fits the past data most closely, but the one that makes the most reliable predictions within a reasonable error margin. Practicing mathematical modeling reinforces critical thinking, statistical analysis, and problem-solving skills – all of which are indispensable in the field of economics.