Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the regression feature of a graphing utility to find an exponential model \(y=a b^{x}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(-3,102.2),(0,80.5),(3,67.8),(6,58.2),(10,55.0)$$

Step-by-Step Solution

Answer

Due to the reliance on a graphing utility for the regression and plotting, the final values of \(a\), \(b\) and \(R^2\) can't be determined without actually running the regression. However, the model will be of the form \(y = ab^x\) and the \(R^2\) value will indicate the fit of this model to the data.

Step 1: Input Data

Enter the given data points into the graphing utility. These are: (-3, 102.2), (0, 80.5), (3, 67.8), (6, 58.2) and (10, 55.0).

Step 2: Run Exponential Regression

Using the available features on the graphing utility, run exponential regression to generate a model in the form of \(y = ab^x\).

Step 3: Identify Model Parameters

Once the regression is complete, the graphing utility will provide the values of \(a\) and \(b\) as well as the coefficient of determination \(R^2\). Note these down.

Step 4: Plot Data and Model

Use the graphing utility to plot the original data points and the fitted model on the same window. This allows us to visualize how well the model represents the given data.

Step 5: Analyze Coefficient of Determination

The coefficient of determination indicates how well the model fits the data. If \(R^2=1\), there is a perfect fit, while \(R^2=0\) indicates no fit. Based on the resulting \(R^2\) value, determine how well the exponential model fits the data.

Key Concepts

Coefficient of Determination

The coefficient of determination, denoted as R-squared (\( R^2 \)), plays a crucial role in understanding the effectiveness of a statistical model, particularly in the context of exponential regression. It quantifies how closely the data points fit the proposed model, essentially measuring the proportion of variance in the dependent variable that's predictable from the independent variable(s).

For example, in an exponential regression of the form \(y=ab^x\), the coefficient of determination gives insight into the accuracy of the model. A \( R^2 \) value of 1 indicates a perfect fit, meaning that the model predicts the data points without any error. Conversely, a value of 0 suggests that the model does not explain any of the variability in the response data around its mean.

When evaluating exponential models, a higher \( R^2 \) value usually suggests a better model fit. However, it's also important to consider other factors such as the complexity of the model and the potential for overfitting, where the model is too closely tailored to the specific dataset and may not perform well on new, unseen data.

Graphing Utility

A graphing utility is a powerful tool used for data plotting and analysis in various fields, including mathematics, engineering, and data science. For educational purposes, it is especially useful for students to visualize mathematical concepts and examine the properties of different functions.

When working with datasets and needing to fit an exponential model, the graphing utility allows for quick and accurate calculations. Tools like these enable students to input data, run regression algorithms like exponential regression, plot the resulting curves, and assess the goodness of fit through metrics such as the coefficient of determination \( R^2 \).

Using a Graphing Utility:

• Enter the provided data points.
• Run functions like regression analysis.
• Observe the calculated model parameters and statistical indices.
• Visualize data and model together on a graph.

Through educational graphing utilities, students can experiment with different types of data and models, sharpening their conceptual understanding and analysis skills.

Exponential Model

An exponential model is often used in statistical analyses to represent data that grows or decays at a rate proportional to its current value. This kind of model is expressed mathematically as \( y = ab^x \), where \( a \) and \( b \) are constants, \( x \) is the independent variable, and \( y \) is the dependent variable.

Whether studying population growth, radioactive decay, or interest compounding in finance, the exponential model provides a framework for analyzing phenomena that exhibit exponential behavior. In an educational context, understanding how to align a set of data points with an exponential curve is paramount.

Characteristics of the Exponential Model:

• It represents changes at a rate proportional to the quantity itself.
• Often used in cases of rapid growth or decay.
• Useful for predictions when the pattern of change is exponential.

By fitting an exponential model to data, students learn not only about the mathematics but also about the interpretable implications of the model in the real world.

Data Plotting

Data plotting is a fundamental aspect of data visualization, which assists in the comprehension and communication of statistical information. In educational settings, it provides a visual means for students to understand the distribution, trends, and patterns within a set of data.

By plotting data points on a coordinate plane, learners can immediately grasp the nature of the data, identifying clusters, gaps, and outliers that might not be evident in a simple list of numbers. In conjunction with fitting models, such as exponential models, data plotting enables the comparison between actual data points and the predictive model.

Benefits of Data Plotting:

• Enhances understanding of the data structure and relationships.
• Fosters visual learning and pattern recognition.
• Facilitates identification of model fit and potential errors.

Importantly, data plotting encourages critical thinking by visually presenting the efficacy of a statistical model, such as confirming the closeness of the exponential fit as indicated by the coefficient of determination.