Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the regression feature of a graphing utility to find a logarithmic model \(y=a+b \ln x\) 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. $$(1, 8.5), (2, 11.4), (4, 12.8), (6, 13.6), (8, 14.2),(10, 14.6)$$

Step-by-Step Solution

Answer

The exact values of coefficients a and b, as well as the coefficient of determination, R-squared, depend on the specific software tool used and their specific algorithms implemented for regression analysis. These cannot be determined without actually running the data through such a tool.

Step 1: Enter the Data

Using the graphing utility, input the given data points: (1, 8.5), (2, 11.4), (4, 12.8), (6, 13.6), (8, 14.2), (10, 14.6)

Step 2: Find the Regression Model

Choose the logarithmic function as your regression model. The graphing utility will produce the equation of the form \(y = a + b \ln x\).

Step 3: Identify the Coefficient of Determination

The coefficient of determination, R-squared, is usually given among the results of the regression model calculation. If R-squared equals 1, then the model perfectly fits the data.

Step 4: Plot the Data and the Model

Now plot the data points on the graph and then superimpose the logarithmic model graph over the data points in the same viewing window. The visual representation will help in understanding the accuracy and reliability of the found model.

Key Concepts

Graphing Utility

A graphing utility is an essential tool that helps us perform complex calculations, such as logarithmic regression, with ease. It is a type of software or calculator that allows you to input data and generate corresponding graphs and models. Graphing utilities can handle a variety of functions and are especially useful in visualizing mathematical equations.

• They enable you to enter data points and plot them on a graph.
• Use the regression feature to find mathematical models like the logarithmic model.
• Graph calculators often include features for various regression types, including linear, quadratic, and logarithmic.

A graphing utility thus simplifies the process of performing statistical analysis and finding best-fit models for data.

Coefficient of Determination

The coefficient of determination, often denoted as R-squared ( ^2"), is a statistic that reveals how well a model fits the data. When we conduct a logarithmic regression, the R-squared value helps us understand the relationship between the variables.

An ^2" value:

• Closer to 1 indicates a strong relationship, meaning the model fits the data well.
• Closer to 0 suggests a weak relationship, with the model not fitting the data precisely.

Knowing the R-squared value can help you assess the accuracy and reliability of your logarithmic model when using a graphing utility.

Plotting Data

Plotting data is a fundamental step in performing regression analysis, as it allows us to visually assess the relationship between variables. In a graphing utility, after entering your data points, plotting them provides an initial overview of your data's trend.

To plot data:

• Input each data point into the graphing utility, ensuring accurate entries.
• View the plotted graph to see the pattern of the data points.
• Determine potential models or functions that might fit the data.

The visual representation through data plotting is crucial as it sets the stage for further analysis, like creating a logarithmic curve.

Logarithmic Model

A logarithmic model is represented by the equation \(y = a + b \ln x\), where:

• \(a\) represents the intercept of the model.
• \(b\) is the coefficient, indicating the slope related to the natural logarithm of the independent variable \(x\).

This type of model is particularly useful when the rate of change of the data decreases rapidly.

When fitting a logarithmic model using a graphing utility, the utility calculates parameters \(a\) and \(b\) that provide the best fit. By graphing this model alongside your data, you can visually confirm how well the model fits, guided by how closely the curve aligns with the data points.
When data follows a logarithmic trend, this kind of regression can effectively capture the underlying pattern, making it a powerful tool in data analysis.