Problem 52 - College Algebra with Corequisite Support

Question

For the following exercises, refer to Table 12 . $$ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} x & 0 & 2 & 4 & 5 & 7 & 8 & 10 & 11 & 15 & 17 \\ \hline f(x) & 12 & 28.6 & 52.8 & 70.3 & 99.9 & 112.5 & 125.8 & 127.9 & 135.1 & 135.9 \end{array} $$ 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

Logistic model constructed with estimated parameters from linear regression.

Step 1: Understanding Logistic Model

We need to fit a logistic growth model to the given data, where the model is of the form \(y=\frac{c}{1+a e^{-b x}}\). The parameters \(a\), \(b\), and \(c\) must be determined to provide the best fit.

Step 2: Initial Parameter Estimation

We start by estimating parameter \(c\), which is often the upper bound of the function or the horizontal asymptote in a real-world context. From the table, as \(x\) increases, \(f(x)\) approaches 135.9. Thus, \(c\) can be approximately taken as 136.

Step 3: Transform Data for Linear Regression

To find \(a\) and \(b\), we linearize the logistic function. Using the identity \(\frac{1}{y} = \frac{1}{c} + \frac{a e^{-bx}}{c}\), rearrange to get \(rac{c - y}{y} = ae^{-bx}\). Applying the natural logarithm, \(\ln(\frac{c-y}{y}) = \ln(a) - bx\). This allows us to use linear regression with \(\ln(\frac{c-y}{y})\) as the dependent variable and \(x\) as the independent variable.

Step 4: Calculate Transformed Values

Calculate \(\ln(\frac{c-f(x)}{f(x)})\) for each pair of \((x, f(x))\) using \(c = 136\). This provides data points to perform linear regression to estimate the parameters \(\ln(a)\) and \(-b\).

Step 5: Perform Linear Regression

Using the transformed data points from Step 4, perform linear regression, which results in an equation of the form \(y = mx + b\). Here, the slope \(m\) gives the estimate for \(-b\), and the y-intercept \(b\) gives \(\ln(a)\).

Step 6: Solve for \(a\) and \(b\)

From the results of the linear regression, use \(b\) (intercept) to find \(a = e^{y-intercept}\) and \(m\) (slope) to find \(b = -m\). This completes the logistic model parameters.

Step 7: Construct the Logistic Model

Insert the calculated parameters \(a\), \(b\), and \(c\) into the logistic model formula \(y=\frac{c}{1+a e^{-b x}}\). This is the finalized logistic model that best fits the given data.

Key Concepts

Data Transformation

Logistic regression involves a specific data transformation process. The idea is to convert a nonlinear problem into a linear one, which is much easier to solve. In our case, the logistic model is expressed as \(y=\frac{c}{1+a e^{-b x}}\). However, this equation is not in a form suitable for linear regression. To linearize it, we start by rearranging: \(\frac{1}{y} = \frac{1}{c} + \frac{a e^{-bx}}{c}\). Then, we simplify it further to \(\frac{c - y}{y} = ae^{-bx}\). This transformation sets the stage for linear regression by allowing us to employ the natural logarithm.Using the natural log on both sides, the equation becomes \(\ln(\frac{c-y}{y}) = \ln(a) - bx\). This is linear with respect to \(x\), where \(\ln(\frac{c-y}{y})\) is the dependent variable. Applying this transformation to the data, we change our nonlinear logistic equation into a format suitable for linear regression analysis.

Linear Regression

Once the data has been transformed, linear regression can be used to identify the model parameters needed for our logistic regression. Linear regression is a method for modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. In the transformed logistic growth model, we treat \(x\) as the independent variable and \(\ln(\frac{c-y}{y})\) as the dependent variable. By performing linear regression, we can determine a line that best fits these transformed data points, described by the equation \(y = mx + b\). Here, \(m\) is the slope of the line, and \(b\) is the y-intercept. These values provide essential information about the parameters of our original logistic growth equation.

Parameter Estimation

The estimated parameters from linear regression help us determine the best fit for our logistic model. First, we estimate parameter \(c\), which is typically the upper bound or the horizontal asymptote of the function. From the table, we observe that as \(x\) increases, \(f(x)\) approaches 135.9. Thus, \(c\) is approximately 136.From the linear regression, we're interested in the slope \(m\) and y-intercept \(b\). The slope is used to find parameter \(-b\) in our logistic model. Negative of the slope gives \(b = -m\). For parameter \(a\), we compute it using the y-intercept \(b\) from linear regression. Specifically, \(a = e^{b}\), with \(e\) being the base of the natural logarithms. Finally, having determined all necessary parameters \(a\), \(b\), and \(c\), we can insert these back into the logistic equation to express the model that best fits the given data.