Problem 25 - 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. $$ \begin{array}{|l|c|c|c|c|c|} \hline \boldsymbol{x} & 8 & 15 & 26 & 31 & 56 \\ \hline \boldsymbol{y} & 23 & 41 & 53 & 72 & 103 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The regression line is \( y = 1.554x + 9.837 \) and \( r = 0.995 \).

Step 1: Enter Data into Calculator

First, enter the given data into your calculator or tool. In the data entry table, assign \( x = [8, 15, 26, 31, 56] \) and \( y = [23, 41, 53, 72, 103] \). Make sure that each \( x \) value corresponds correctly to each \( y \) value.

Step 2: Use Linear Regression Function

Next, use the linear regression function on your calculator or tool. This function usually is accessed via the 'Stat' button followed by 'Calc' and choosing 'LinReg'. Select the lists where you have entered \( x \) and \( y \) values, and execute the function to find the regression line.

Step 3: Determine Equation of Regression Line

The tool will output the parameters \( a \) and \( b \) for the line of best fit. These parameters represent the intercept and slope of the line, respectively. The regression line equation is given as \( y = ax + b \). For this data set, it calculates to be \( y = 1.554x + 9.837 \).

Step 4: Calculate and Note Correlation Coefficient

The tool will also provide the correlation coefficient \( r \), which indicates the strength and direction of the relationship between \( x \) and \( y \). In this case, the correlation coefficient is approximately \( r = 0.995 \).

Step 5: Interpret the Results

The linear equation \( y = 1.554x + 9.837 \) suggests that for each unit increase in \( x \), \( y \) is expected to increase by approximately 1.554 units. A correlation coefficient of \( r = 0.995 \) signifies a very strong positive linear relationship between \( x \) and \( y \).

Key Concepts

Understanding the Correlation Coefficient

The correlation coefficient, represented as \( r \), is a statistic used to measure the strength and direction of a linear relationship between two variables. It ranges from \(-1\) to \(1\).

• An \( r \) value of \(1\) indicates a perfect positive linear relationship, meaning as one variable increases, the other increases proportionally.
• An \( r \) value of \(-1\) indicates a perfect negative linear relationship, where as one increases, the other decreases proportionally.
• An \( r \) value close to \(0\) suggests little or no linear relationship between the variables.

The exercise discusses a correlation coefficient of \( r = 0.995 \). This means there's an almost perfect positive relationship. So as \( x \) increases, \( y \) tends to increase significantly.
High correlation values, like \(0.995\), are often sought in predictive data scenarios as they indicate reliable predictions.

Explaining Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In simple linear regression, which involves two variables, the aim is to find the best-fit line, described by \( y = ax + b \).

How It Works

• \( a \) is the slope, which shows how much \( y \) changes for a unit change in \( x \).
• \( b \) is the y-intercept, representing the value of \( y \) when \( x \) is zero.

From the solution, we get \( y = 1.554x + 9.837 \). This equation tells us that for every unit increase in \( x \), the predicted \( y \) value increases by \(1.554\) units.
This line of best fit is instrumental in forecasting and interpreting patterns in data, making linear regression a powerful tool in data analysis.

The Role of Data Analysis

Data analysis involves examining, organizing, and interpreting data to extract valuable insights. In the context of regression, it helps us understand relationships and predict future outcomes.

Steps in Data Analysis

1. **Data Collection**: Gather and input data into a system or tool.
2. **Data Interpretation**: Use tools like linear regression to comprehend relationships.
3. **Conclusion Drawing**: Interpret results and understand implications. For instance, strong correlation suggests reliable data predictions.
Learning to use these techniques gives you the ability to handle real-world data, leading to informed decision-making based on observed trends.
This exercise shows how crucial accurate analysis is. It not only provides the regression line and correlation coefficient but also translates theoretical results into practical insights.