Problem 21 - 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. $$(0,8.3),(1,6.1),(2,4.6),(3,3.8),(4,3.6)$$

Step-by-Step Solution

Answer

Values for \(a\) and \(b\) will differ depending on the specific calculator used, so a direct answer can't be provided. However, the coefficient of determination, \(r^2\), and the visualization of the data points and the model graph will show how well the exponential model fits the given data.

Step 1: Insert the data into the graphing utility

Firstly, insert the given data pairs into the calculator. The first number in each pair will be the \(x\) value, and the second number will be the \(y\) value. The given data pairs are (0,8.3), (1,6.1), (2,4.6), (3,3.8), and (4,3.6).

Step 2: Perform the exponential regression and find the model equation

Then, use the graphing utility's functionality to perform an exponential regression. The calculator will provide values for \(a\) and \(b\) in the equation \(y=ab^x\).

Step 3: Identify the coefficient of determination

The graphing utility should also provide the coefficient of determination, usually denoted as \(r^2\), which tells how well the exponential model fits the data. The closer the \(r^2\) value is to 1, the better the model fits the data.

Step 4: Plot the data and graph the model on the same viewing window

Finally, plot the data points on a graph, and then graph the regression model equation on the same window. This will allow you to visually compare how well the model fits the data.

Key Concepts

graphing utility

A graphing utility can be a powerful tool when it comes to understanding and visualizing mathematical data. Imagine it as a super calculator that does more than just basic arithmetic. It helps to visualize how different values relate to each other by plotting them on a graph.
Bringing numbers into a picture makes it much easier to comprehend complex relationships. Many graphing utilities have features like regression analysis, which assists in finding equations that best describe your data. In our case, it helps find the best exponential model for a given data set.
To begin using a graphing utility, input your data points, like:

• (0,8.3)
• (1,6.1)
• (2,4.6)
• (3,3.8)
• (4,3.6)

Once the data is in, the graphing utility can find patterns or trends, allowing you to perform further calculations like exponential regression.

coefficient of determination

The coefficient of determination is a statistical measure denoted as \(r^2\) and is crucial in regression analysis. It tells us how well a model fits the data; essentially, it measures the accuracy of the model.
If we've got a perfect fit, \(r^2\) will equal 1, meaning 100% of the variation in the data is explained by the model. However, if \(r^2\) is closer to 0, it indicates that the model does not represent the data well.
To find the coefficient of determination for an exponential model in a graphing utility, you perform regression on the data. The calculator will provide this value automatically. Remember, a higher \(r^2\) value signifies a more reliable model equation.

plotting data

Plotting data is the initial step in visually analyzing the relationship between variables in a data set. It involves placing data points on a coordinate plane where each point corresponds to a pair of values. For example, the point (0,8.3) means that at \(x = 0\), \(y = 8.3\).
When you plot all your data points:

• You can easily see trends.
• You can identify outliers or data inconsistencies.
• It sets the stage for further analysis with regression fits.

In our case, after plotting, you can see if the points align with an exponential curve, which would justify using exponential regression as the appropriate model.

model equation

The model equation derived from exponential regression represents the relationship between the variables in the data. In this context, the form is \(y = ab^x\), where \(a\) is the initial value when \(x = 0\), and \(b\) is the growth factor, determining how \(y\) changes as \(x\) increases.
Once you input the data into the graphing utility and request it to perform exponential regression, it provides the values for \(a\) and \(b\) in the equation. Suppose it returns \(a = 8\) and \(b = 0.7\). Therefore, your model equation becomes \(y = 8 \times 0.7^x\).
This equation can now be graphed to check how well it fits the actual data points, guided by your coefficient of determination \(r^2\) value. The closer the points lie to this graph, the better the model describes your data.