Problem 26 - 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. $$ \begin{array}{|l|l|l|l|l|l|} \hline x & 5 & 7 & 10 & 12 & 15 \\ \hline y & 4 & 12 & 17 & 22 & 24 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The regression line is approximately \(y = 1.536x + 1.32\) with a correlation coefficient of \(0.965\).

Step 1: Input Data

Enter the given data points into the calculator or software. You will input two lists or columns of data: one for \(x\) values (5, 7, 10, 12, 15) and one for \(y\) values (4, 12, 17, 22, 24).

Step 2: Use Regression Function

Use the regression function provided by the calculator or software tool to compute the regression line. Most tools have a linear regression feature, often denoted as LinReg, which calculates the regression line \(y = ax + b\).

Step 3: Obtain Regression Equation

After calculating the regression line, note down the equation of the line. Based on the input data, the calculator will output the values of \(a\) (slope) and \(b\) (y-intercept) for the equation \(y = ax + b\).

Step 4: Compute the Correlation Coefficient

The calculator or the software will also provide the correlation coefficient, \(r\), which indicates the strength and direction of the linear relationship between \(x\) and \(y\). Ensure that \(r\) is recorded up to three decimal places.

Key Concepts

Understanding the Correlation Coefficient

The correlation coefficient, often represented by the letter \(r\), is a crucial number in statistics that measures the strength and direction of a linear relationship between two variables. In our exercise, we're looking at the data pairs \((x, y)\) to understand their correlation.

• If \(r\) is close to 1, it means there's a strong positive linear relationship. As \(x\) increases, \(y\) tends to increase as well.
• If \(r\) is close to -1, the relationship is strong but negative. Here, as \(x\) goes up, \(y\) tends to go down.
• If \(r\) is around 0, it suggests no linear relationship between the variables.

Computing \(r\) accurately tells us a lot about the predictive value of our \(x\) and \(y\) data. For this exercise, computing \(r\) helps verify the reliability of the linear regression model we create.

Constructing the Regression Equation

The regression equation is an essential tool in statistics. It provides a mathematical method to predict or describe the relationship between two variables. For our dataset, we use the linear regression method to compute this equation, which commonly looks like \(y = ax + b\):

• \(a\) is the slope of the line; it represents the rate of change between the dependent and independent variables.
• \(b\) is the y-intercept; the point where the line crosses the y-axis when \(x = 0\).

By inputting the \(x\) and \(y\) values into a calculator's regression function, you obtain \(a\) and \(b\). For our exercise, these values are critical as they enable us to predict the \(y\) value for any \(x\) in the given range. Thus, the regression equation not only models the data but helps us make predictions about trends.

Effective Data Analysis

Data analysis is a systematic process of inspecting and modeling data with the end goal of discovering useful information. In our exercise involving data pairs \((x, y)\), effective data analysis involves several key steps:

• Entering data properly: Ensure that data is entered accurately in your calculator or software to avoid errors in the results.
• Using technology tools: Choose reliable technology tools like calculators or software that support regression functions.
• Interpreting results: Once the correlation coefficient and regression equation are computed, interpret these results to derive meaningful insights about the dataset and relationships between variables.

Data analysis involves looking at the variability and trends shown in our results. This process not only involves calculating figures but understanding what these results mean in the context of the study objectives. When done well, it helps us make data-driven predictions and decisions.