Problem 25 - 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. $$(1,11),(2,6),(3,5),(4,4),(5,3),(6,2)$$

Step-by-Step Solution

Answer

The exact answer depends on the results from the graphing utility used to perform the regression. The logarithmic model will be in the form \(y = a + b \ln x\), and the coefficient of determination \(R^2\) will reveal how precisely the model fits the data.

Step 1: Input the Data Points

In the graphing utility, input the data points given: (1,11),(2,6),(3,5),(4,4),(5,3),(6,2).

Step 2: Perform the Logarithmic Regression

Use the regression feature within the utility to calculate the logarithmic regression. This should output a function in the form \(y = a + b \ln x \), where 'a' and 'b' are coefficients determined by the program.

Step 3: Identify the Logarithmic Model and Coefficient of Determination

The calculated logarithmic regression function generated by the utility should be of the form \(y = a + b \ln x\), where 'a' and 'b' are the coefficients obtained. These coefficients help us understand the behavior of the data. The utility should also provide an indication of the coefficient of determination \(R^2\). This coefficient gives us an idea of how well the model fits the data.

Step 4: Plot Data and Model

Plot the provided data points and the calculated regression model in the same graphing window. One should be able to visually verify how well the model fits the given data.

Key Concepts

Graphing Utility

A graphing utility is a vital tool for students and professionals working with mathematics and statistics. It allows users to input data points and visualize how different types of models fit those points. In this exercise, the graphing utility plays a crucial role in performing a logarithmic regression. By entering the data points (1, 11), (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2) into the tool, you can use its regression feature to calculate a logarithmic model.

Graphing utilities provide an intuitive way to see your data and regression model simultaneously. This is done by plotting both on the same graph, helping users visually assess how well the model aligns with the data. Many graphing calculators or software applications offer various features, such as zooming in on graphs, adjusting viewing windows, and even overlaying different types of regression models.

• Make sure to correctly input each data point.
• Use the built-in regression function to fit the model.
• View both the data and the fitted model together.

Coefficient of Determination

The coefficient of determination, often represented as \(R^2\), is a statistical measure that provides insights into the goodness of fit of a regression model. In simpler terms, it helps you understand how well the chosen model explains the variability of the data.

In the context of a logarithmic regression, the coefficient of determination ranges from 0 to 1. An \(R^2\) value closer to 1 indicates that the model fits the data well, capturing much of the variability. Conversely, a value closer to 0 suggests that the model does not fit the data well, capturing little of the variability.

• \(R^2 = 1\) suggests a perfect fit.
• \(R^2 = 0\) suggests no fit at all.
• Provides a numerical value for model fitting quality.

When you perform a logarithmic regression using a graphing utility, it will often display \(R^2\) automatically. This makes it easy to assess how successful your model is in representing the data.

Logarithmic Model

A logarithmic model is one type of regression model useful in statistical analysis, particularly when dealing with data showing a slower rate of increase as time progresses.

The model is formulated as \(y = a + b \ln x\), where:

• \(a\) is the y-intercept, reflecting the value of \(y\) when \(x\) is 1.
• \(b\) is the slope, indicating the rate of change relative to the natural logarithm of \(x\).
• \(\ln x\) is the natural logarithm of \(x\).

In this exercise, logarithmic models help represent relationships where variables change at a decreasing rate. These models are particularly helpful for fitting data where the rate of change decreases rapidly at first, then slows down. With the graphing utility, you can quickly derive coefficient values for \(a\) and \(b\), providing you with a mathematical equation that best fits your data.