Problem 27 - College Algebra with Corequisite Support

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. $$ \begin{array}{|c|c|c|c|} \hline x & y & x & y \\ \hline 3 & 21.9 & 10 & 18.54 \\ \hline 4 & 22.22 & 11 & 15.76 \\ \hline 5 & 22.74 & 12 & 13.68 \\ \hline 6 & 22.26 & 13 & 14.1 \\ \hline 7 & 20.78 & 14 & 14.02 \\ \hline 8 & 17.6 & 15 & 11.94 \\ \hline 9 & 16.52 & 16 & 12.76 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The regression line is \(y = -0.497x + 22.5\) and the correlation coefficient is \(-0.608\).

Step 1: Organize the Data

To simplify the calculation, arrange the data into two separate lists: one for \(x\) and one for \(y\). For this dataset: \[ x = [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] \] \[ y = [21.9, 22.22, 22.74, 22.26, 20.78, 17.6, 16.52, 18.54, 15.76, 13.68, 14.1, 14.02, 11.94, 12.76] \]

Step 2: Input Data into a Calculator or Software

Use a calculator or spreadsheet software to input the \(x\) and \(y\) data points. Ensure that each \(x\) value is paired with its corresponding \(y\) value.

Step 3: Calculate the Regression Line

Apply the linear regression function available in the calculator or software to the data points. This provides the equation of the line in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For this dataset: \(m \approx -0.497\), \(b \approx 22.5\). Thus, the regression line is \(y = -0.497x + 22.5\).

Step 4: Determine the Correlation Coefficient

Using the same calculator or software, find the correlation coefficient \(r\), which measures the strength and direction of the linear relationship between \(x\) and \(y\). For these data points, \(r \approx -0.608\).

Step 5: Round the Correlation Coefficient

Round the correlation coefficient to three decimal places for final reporting. The rounded value is \(-0.608\).

Key Concepts

Linear Regression

Linear regression is a fundamental data analysis tool. It's used to explore the relationship between two variables, typically denoted as \(x\) and \(y\). The goal is to find an equation of the form \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. This equation models how the variable \(y\) changes concerning \(x\).
A simple example is predicting someone's income based on their years of education, where education is \(x\), and income is \(y\). Linear regression will help draw the best-fit line through the plotted data points. This line minimizes the distance, called the residuals, between the observed values and the values predicted by the line.
The values of \(m\) and \(b\) are calculated such that the sum of squared residuals is minimized. The smaller the residuals, the better the fit of the line to the data. This makes linear regression highly valuable for making predictions and understanding relationships in datasets.

Correlation Coefficient

The correlation coefficient, denoted as \(r\), quantifies the degree to which two variables are related. This metric ranges from -1 to 1. A value close to 1 indicates a strong positive relationship, meaning as one variable increases, the other also does. Conversely, a value close to -1 indicates a strong negative relationship. Here, as one variable increases, the other decreases.
If \(r\) is around 0, it suggests no linear relationship between the variables. In our example, the calculated \(r\) was approximately \(-0.608\). This implies a moderate, negative linear relationship between the \(x\) and \(y\) datasets. Practically, this means that as \(x\) increases, \(y\) tends to decrease, but not in a perfectly linear manner.
Understanding \(r\) helps in assessing the strength and direction of relationships, which can be crucial for data analysis, trend forecasting, and hypothesis testing.

Data Analysis

Data analysis is a process that helps you make sense of raw data. By using statistical techniques, such as linear regression, you can uncover trends, patterns, and insights from data.
In the context of the given exercise, data analysis started with organizing the data into lists of \(x\) and \(y\) values. This setup was vital for inputting into a calculator or software for further exploration. It allowed for efficient and error-free computation of the regression line and correlation coefficient.

• First, clean and organize data to ensure accuracy in subsequent steps.
• Utilize technology tools, like calculators and spreadsheet software, which provide access to functions for complex calculations.
• Interpret results to draw meaningful conclusions that can influence decisions or guide further research.

Through data analysis, you gain the ability to predict future values and better understand the interplay between different variables. This skill is invaluable in fields ranging from economics to biology, wherever data-driven decision making is crucial.