Problem 35 - Calculus an Applied Approach

Question

In Exercises \(35-38,\) use the regression capabilities of a graphing utility or a spreadsheet to find any model that best fits the data points. $$ \begin{array}{l}{(1,13),(2,16.5),(4,24),(5,28),(8,39),(11,50.25)} \\\ {(17,72),(20,85)}\end{array} $$

Step-by-Step Solution

Answer

Since the software or utility used for the regression analysis is not specified, we cannot provide a specific answer. It is important to note that while running the linear regression analysis, the software will provide a linear model equation in the form \(y = mx + c\) where m is the slope and c is the y-intercept. This equation best fits the given data.

Step1: Arrange Data

Firstly, arrange the given data points in a two columns: one for x-values and one for y-values. This can be done either on a graphing calculator or a spreadsheet software such as Excel.

Step 2: Perform Linear Regression

Tools such as Microsoft Excel or a graphing calculator provide the function to fit a curve, specifically a line in this case, to the given data set. Run linear regression on the data set to find the model of the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Step 3: Determine the Linear Equation

From the result of the linear regression operation, extract values of \(m\) and \(c\) to establish your linear equation that best represents the data points.

Step 4: Test the Fit

To verify if the obtained model sufficiently fits the data, it's useful to plot the given data points and the resulting regression line on the same graph. This visual approach makes it easier to assess how well the model represents the actual data.

Key Concepts

Graphing Utility

A graphing utility is a valuable tool used to visualize mathematical functions and data sets. These tools can create graphs that provide insight into data trends and relationships between variables.
Graphing utilities can be physical calculators or software applications that simulate a calculator's function on a computer or smartphone.
An essential feature of graphing utilities is their ability to perform regression analysis. They help find the best-fit line or curve for a set of data points, making it easier to see patterns at a glance. When dealing with exercises that involve finding a model for data points, a graphing utility allows users to execute and visualize the fitting process efficiently.

Data Fitting

Data fitting refers to the process of finding a mathematical model that best captures the relationship between a set of data points. In linear regression, this involves identifying a line that minimizes the overall difference between the data points and the line itself.
The best-fit line is characterized by its slope and y-intercept, forming an equation of the type:

• Slope (\(m\)) indicates the steepness of the line and the direction of the relationship.
• Y-intercept (\(c\)) represents the point where the line crosses the y-axis.

Performing data fitting allows us to make predictions, understand trends, and derive conclusions about the data set. By applying linear regression, we aim to reduce the discrepancies between the predicted values and the actual data values as much as possible.

Linear Equation

A linear equation is a mathematical expression that creates a straight line when plotted on a graph. It is typically written in the form:\[ y = mx + c \] where:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• \(m\) is the slope; it determines the line's angle.
• \(c\) is the y-intercept; it is the starting point of the line on the y-axis.

In the context of data sets, a linear equation is often used to model the relationship between two variables. When we perform linear regression on a set of data points, we arrive at a linear equation that best fits those points. This equation can then be used to make predictions or understand the nature of the data.

Spreadsheet Software

Spreadsheet software, like Microsoft Excel or Google Sheets, is a powerful tool for handling and analyzing data. These programs offer extensive features for data organization, calculation, and graphing.
One of the key capabilities of spreadsheet software is performing regression analysis, such as linear regression, on data sets. They excel in creating graphs that help visualize data fitting results, making it easier to interpret the outcome of statistical analyses.
Users can input data directly into spreadsheet cells, and with a few commands or clicks, they can generate a linear model and plot both the data points and regression line on a chart. This process streamlines the approach to data analysis, making it accessible even for beginners to understand and apply.