Problem 30 - College Algebra

Question

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 100 & 80 & 60 & 55 & 40 & 20 \\ \hline \boldsymbol{y} & 2000 & 1798 & 1589 & 1580 & 1390 & 1202 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Regression line: \( y = -15.35x + 3386.4 \), correlation coefficient: \(-0.998\).

Step 1: Organize the Data

List the data in two separate sequences: one for \(x\) values and another for \(y\) values. Here they are: \(x = [100, 80, 60, 55, 40, 20]\) and \(y = [2000, 1798, 1589, 1580, 1390, 1202]\).

Step 2: Input Data into a Calculator

Most scientific calculators or spreadsheet software can calculate regression lines. Input the \(x\) and \(y\) data into the calculator's regression function or a spreadsheet.

Step 3: Calculate the Regression Line

Use the tool to compute the line of best fit, which is often in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The result will provide the specific numerical values for \(m\) and \(b\).

Step 4: Extract the Coefficients

Identify the slope \(m\) and y-intercept \(b\) from the regression output. For this dataset, the regression line may look like \(y = -15.35x + 3386.4\) based on an external calculation tool.

Step 5: Determine the Correlation Coefficient

Most calculators or software also give the correlation coefficient \(r\), which is a measure of the strength and direction of the relationship. This value should be between -1 and 1. For this dataset, \(r\approx -0.998\).

Step 6: Interpret the Results

The regression line equation indicates a strong negative linear relationship, as shown by the correlation coefficient close to \(-1\). This suggests that as \(x\) increases, \(y\) decreases predictably.

Key Concepts

Regression Line

When dealing with data sets, one often encounters the goal of making predictions about future or unseen data points. This is where the concept of a regression line comes into play. The regression line, often known as the line of best fit, provides a simple linear equation that best describes the relationship between a dependent variable, typically represented as \( y \), and an independent variable, represented as \( x \). This relationship is expressed in the linear equation form \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating how much \( y \) changes for a unit change in \( x \).
• \( b \) is the y-intercept, which is the value of \( y \) when \( x = 0 \).

Creating a regression line gives us a visual and mathematical tool for prediction, helping us understand how significant the changes in \( x \) will affect \( y \). In practical applications, the regression line can be graphically plotted on a scatter plot of the data, reinforcing how well our line fits through the points.

Correlation Coefficient

The correlation coefficient, often denoted as \( r \), offers insight into how strongly two variables are related. Ranging from -1 to 1, it not only indicates the strength but also the direction of a linear relationship between variables.

• A correlation coefficient of \( 1 \) signifies a perfect positive linear relationship, meaning as \( x \) increases, \( y \) increases.
• A coefficient of \(-1\) implies a perfect negative linear relationship; as \( x \) increases, \( y \) decreases.
• Values close to 0 suggest a weak or no linear relationship between variables.

For example, in the provided dataset, the correlation coefficient is approximately \(-0.998\), suggesting a very strong negative relationship. This indicates that as the \( x \) values increase, the \( y \) values tend to decrease significantly, almost in a predictable manner. Understanding \( r \) helps clarify the predictive power and reliability of your regression line.

Calculator Use

In the realm of statistics, technology serves as a powerful ally, significantly easing the computation of complex equations and data manipulations. Calculating a regression line and a correlation coefficient manually could be daunting, but a scientific calculator or spreadsheet software like Excel can streamline this process.
Steps to use a calculator typically involve:

• Entering the \( x \) and \( y \) data points into a dedicated function for regression calculations.
• Utilizing the calculator's "regression" function to compute the slope \( m \) and intercept \( b \), as well as the correlation coefficient \( r \).
• Reviewing the output to ensure all values make sense concerning your data.

This process can vary slightly depending on the calculator model or software, but the core idea remains the same. Familiarizing yourself with these tools is beneficial, as they simplify the process of identifying trends and making predictions.

Data Organization

Before diving into calculations, organizing your data is a fundamental step that should not be overlooked. Proper data organization aids in precision and allows for an efficient and error-free analysis process.

• Start by listing out all \( x \) values in a sequence, ensuring none are missing or duplicated unless repeated intentionally.
• Correspondingly, list all \( y \) values, aligning them correctly with their respective \( x \) counterparts.

It's often helpful to set up a simple table or spreadsheet where each row represents a pair of \( x \) and \( y \) values. This setup will serve you well, particularly when entering data into a calculator or software for regression analysis. Keeping your data organized ensures that calculations run smoothly and results are accurate, providing a reliable foundation for interpreting your findings and drawing relevant conclusions.