Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, refer to \(\underline{\text { Table } 11 .}\) $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 8.7 & 12.3 & 15.4 & 18.5 & 20.7 & 22.5 & 23.3 & 24 & 24.6 & 24.8 \\ \hline \end{array} $$ Use the LOGISTIC regression option to find a logistic growth model of the form \(y=\frac{c}{1+a e^{-b x}}\) that best fits the data in the table.

Step-by-Step Solution

Answer

Use logistic regression software to determine the parameters \( a, b, \) and \( c \) for the model \( y=\frac{c}{1+a e^{-b x}} \).

Step 1: Understand the Exercise

We need to find a logistic growth model for the given data using the logistic regression option. The form of the model is \( y=\frac{c}{1+a e^{-b x}} \).

Step 2: Identify the Variables

The variables provided in the table are \( x \), which represents the independent variable, and \( f(x) \), which represents the dependent variable. These two sets of numbers will be used to fit the logistic model.

Step 3: Perform Logistic Regression

Logistic regression is typically performed using software or a calculator that supports this feature. We input the \( x \) values and the corresponding \( f(x) \) values to get the parameters \( a, b, \) and \( c \).

Step 4: Interpret the Regression Output

After processing the data with logistic regression software, you will obtain the values for \( a, b, \) and \( c \) to create the logistic model. An example output might be \( a = 2.5 \), \( b = 0.7 \), and \( c = 24.8 \), though these values will depend on the software output.

Step 5: Construct the Logistic Model

Using the obtained parameters, substitute into the logistic model formula: \( y=\frac{c}{1+a e^{-b x}} \). For example, if \( a = 2.5 \), \( b = 0.7 \), and \( c = 24.8 \), the model is \( y=\frac{24.8}{1+2.5 e^{-0.7 x}} \).

Step 6: Verify the Model's Fit

Evaluate the logistic function with the derived parameters over the data points to ensure that it models the data well. Compare the values of the logistic model to \( f(x) \) from the data to confirm fit quality.

Key Concepts

Logistic Growth Model

A logistic growth model is a type of mathematical function that describes how a quantity grows rapidly at first and then slows down, finally settling into a stable state. This is particularly useful for modeling real-life situations where growth starts exponentially, like population growth or the spread of diseases, but eventually levels off due to constraints such as limited resources. In the context of logistic regression, we use a specific mathematical form, which can be expressed as:
\[y = \frac{c}{1 + a e^{-b x}}\]
This equation adjusts the growth so that initially, the growth doubles rapidly but slows as the variable approaches its carrying capacity \(c\), the theoretical maximum value.

• \(c\): The carrying capacity or the maximum value the dependent variable can take.
• \(a\), \(b\): Constants that affect the rate and the shift of the growth curve.

Such models are useful in fields ranging from biology to economics, where the saturation point is significant.

Independent Variable

The independent variable in a dataset is the variable that is manipulated or can be changed and determines the result of a dependent variable. In our logistic growth model, the independent variable is represented by \(x\). It is often the input or the factor you expect to influence the outcome, which in this case is the dependent variable \(f(x)\). When crafting models:

• Choose independent variables based on relevance.
• Ensure they are measurable and can affect the dependent variable.
• Consider if the data is already in a format suitable for analysis.

In our exercise, each value of \(x\) from the table has a corresponding \(f(x)\), signifying that changes in \(x\) are expected to explain the variations in \(f(x)\). Understanding the role of \(x\) as an independent variable helps determine its influence on the modeled outcome.

Dependent Variable

The dependent variable is what we measure in an experiment or study and what is affected during the experiment. It's dependent on the independent variable, linked in our logistic growth model as \(f(x)\). This represents the observed outcomes that the model aims to predict or explain based on changes to \(x\).
The dependent variable should reflect the change as the independent variable is manipulated. In our example, as \(x\) increases, \(f(x)\) starts small and increases, then begins to plateau, demonstrating a classic logistic growth scenario. Here are some key points to consider about dependent variables:

• It should be directly measurable and accurately represent the outcome of interest.
• Its variation provides insights into the relationship being studied.
• It is used to test the hypotheses regarding the impact of the independent variable.

Recognizing \(f(x)\) as a dependent variable is key for understanding how it dynamically changes with \(x\), following the logistical pattern of growth.

Model Fitting

Model fitting is the process of determining the best parameters \(a\), \(b\), and \(c\) for our logistic growth model equation that closely captures the observed data in the table. This involves using statistical techniques to adjust the model so that its output matches the given data as closely as possible.
The goal of model fitting is to minimize discrepancies between observed and predicted data points by finding optimal values that adjust the logistic curve accurately. This is often achieved through:

• Iterative processes provided by statistical software or tools.
• Optimizing a fit criterion, typically reducing the sum of squared differences between the observed and model-predicted values.
• Assessing the goodness of fit through metrics like R-squared values or residual analysis.

Effective model fitting ensures that our logistic model not only fits the current observed data well but is also capable of making accurate predictions on new data. This is a crucial step in making the model a useful tool for simulation and predictive analyses.