What Is The Exponential Regression Equation That Fits These Data Title: “Utilizing Regression Analysis to Create Mathematical Models from Data”
Learning Objectives In this section, you will:
- Develop exponential models from data.
- Create logarithmic models from data.
- Construct logistic models from data.
In the preceding sections of this chapter, we were presented with functions to graph or evaluate explicitly, or we were supplied with a set of points guaranteed to lie on the curve. We employed algebraic methods to discover the equation that precisely matched those points. However, in this section, we employ a modeling approach known as regression analysis to derive a curve that represents data gathered from real-world observations. In regression analysis, we do not anticipate all data points to align perfectly with the curve. The goal is to find a model that best fits the data, which can then be used to predict future events.
Do not be perplexed by the term “model.” In mathematics, we often use the terms “function,” “equation,” and “model” interchangeably, despite each having its distinct formal definition. The term “model” generally implies that the equation or function approximates a real-world scenario.
We will focus on three types of regression models in this section: exponential, logarithmic, and logistic. Our previous experience with each of these functions provides us with an advantage. Understanding their formal definitions, the characteristics of their graphs, and some of their real-world applications allows us to deepen our comprehension. As each regression model is introduced, we will include key features and definitions related to its associated function. Take a moment to revisit each of these functions, reflect on our previous work, and then explore how regression is applied to model real-world phenomena.
Building an Exponential Model from Data
We have learned that exponential functions can model various scenarios, such as investment growth, radioactive decay, atmospheric pressure changes, and the cooling of objects. What do these phenomena have in common? Firstly, all these models either increase or decrease as time progresses. However, there’s more to it. It’s the manner in which the data increase or decrease that helps us determine if an exponential equation is the best fit. Let’s review exponential growth and decay.
Recall that exponential functions have the forms:
- y=abxy=abx or 2. y=A0ekxy=A0ekx.
In regression analysis, we predominantly use the form y=abxy=abx. Reflect on the properties we’ve already learned about the exponential function y=abxy=abx (assuming a>0a>0):
- bb must be greater than zero and not equal to one.
- The initial value of the model is y=ay=a.
- If b>1b>1, the function models exponential growth. As xx increases, the outputs of the model increase slowly at first, but then increase more and more rapidly without bound.
- If 0<b<10<b<1, the function models exponential decay. As xx increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the x-axis. In other words, the outputs never become equal to or less than zero.
When performing regression analysis, your calculator will display a number known as the correlation coefficient, often labeled rr or r2r2 (you may need to adjust the calculator’s settings to display these). These values indicate the “goodness of fit” of the regression equation to the data. We more commonly use the value of r2r2 instead of rr, but the closer either value is to 1, the better the regression equation approximates the data.
Exponential regression is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to approach zero. To fit an exponential function to a set of data points, we utilize the “ExpReg” command on a graphing utility. This returns an equation in the form:
- bb must be non-negative.
- When b>1b>1, we have an exponential growth model.
- When 0<b<10<b<1, we have an exponential decay model.
To perform exponential regression using a graphing utility:
- Use the STAT then EDIT menu to enter the given data.
- Clear any existing data from the lists.
- List the input values in the L1 column.
- List the output values in the L2 column.
- Graph and observe a scatter plot of the data using the STATPLOT feature.
- Use ZOOM  to adjust axes to fit the data.
- Verify that the data exhibit an exponential pattern.
- Find the equation that models the data.
- Select “ExpReg” from the STAT then CALC menu.
- Record the values returned for aa and bb to represent the model y=abxy=abx.
- Graph the model alongside the scatterplot to confirm its suitability for the data.
Using Exponential Regression to Fit a Model to Data:
In 2007, a university study was published investigating the crash risk of alcohol-impaired driving. Data from 2,871 crashes were used to measure the association between a person’s blood alcohol level