Problem 31 - 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. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 900 & 988 & 1000 & 1010 & 1200 & 1205 \\ \hline \boldsymbol{y} & 70 & 80 & 82 & 84 & 105 & 108 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The regression line is \(y = 0.093x - 10.142\), and the correlation coefficient is 0.997.

Step 1: Enter Data into the Calculator

Use a graphing calculator or statistical software to input the given data sets. Input the values of \(x\) and \(y\) into two separate lists. For this example, List 1 (L1) will contain the \(x\)-values: [900, 988, 1000, 1010, 1200, 1205] and List 2 (L2) will contain the \(y\)-values: [70, 80, 82, 84, 105, 108].

Step 2: Calculate the Regression Line

Utilize the regression function on your calculator or software to compute the linear regression line. Generally, navigate to the 'Stat' menu, select 'Calculate', and then choose 'LinReg(ax+b)' or equivalent to calculate. Apply the function with the lists L1 and L2. The output will give the slope \(a\) and the y-intercept \(b\) of the regression line.

Step 3: Retrieve Regression Equation

Based on the calculator's output, write down the linear regression equation in the form \(y = ax + b\). For example, if the calculator indicates that \(a = 0.093\) and \(b = -10.142\), then the equation would be \(y = 0.093x - 10.142\).

Step 4: Find the Correlation Coefficient

Check the output from the regression calculation for the correlation coefficient \(r\), which measures the strength and direction of the linear relationship between \(x\) and \(y\). Record this value to 3 decimal places. For instance, if the calculator outputs \(r = 0.997\), note \(r = 0.997\).

Step 5: Interpret the Results

With the equation and the correlation coefficient, interpret what these mean. A high \(r\) value close to 1 indicates a strong positive linear relationship. Ensure clarity on how these results relate to the data set.

Key Concepts

Regression Line Calculation

When tackling a regression line calculation, the main goal is to accurately depict the relationship between two variables using a straight line. This line, often referred to as the "best fit" line, simplifies the process of making predictions based on the data provided.

• The starting point involves gathering your data, which includes pairs of x and y values.
• Using a calculator or software, input these values into two lists for easy visualization and computation.
• Access the regression function, typically found in the 'Stat' menu, and execute the linear regression calculation.

You will obtain two key values: the slope (a) and the intercept (b). These are used to formulate the regression equation in the form: \(y = ax + b\). The slope represents the rate of change of y with respect to x, essentially indicating whether the relationship is positive or negative. Meanwhile, the y-intercept denotes the point at which the line crosses the y-axis, offering insights into the value of y when x equals zero.

Correlation Coefficient

The correlation coefficient, often symbolized as \(r\), quantifies the musical harmony between your variables. It's a measure of the strength and direction of the linear relationship between them.

• A value of \(r\) close to 1 suggests a strong positive relationship, where increases in x typically lead to increases in y.
• Conversely, a value close to -1 implies a strong negative relationship, indicating that increases in one variable correspond to decreases in the other.
• If \(r\) is around zero, this implies little to no linear relationship between the variables.

Calculating \(r\) involves using built-in functions on calculators or statistical software, which quickly output this value after entering your data. For example, in the given data, if \(r = 0.997\), you can infer a nearly perfect positive linear relationship.

Graphing Calculator Usage

Graphing calculators are powerful tools for understanding and visualizing data relationships. They expedite complex computations and can display both algebraic and graphical results.

• Start by entering the x-values and y-values into two separate lists, often labeled L1 and L2.
• Navigate to the calculation section, typically under the 'Stat' menu, and select the 'LinReg(ax+b)' function.
• Input the respective lists and run the function to calculate the regression line and the correlation coefficient.

Graphical capabilities of these calculators often allow for the plotted regression line to be seen in real time on a Cartesian plane, offering a visual confirmation of the linear relationship.

Linear Relationship Interpretation

Understanding linear relationships allows you to make educated predictions and analyze trends in datasets. Interpretation is key to applying statistical findings to real-world scenarios.

• With a determined regression equation, interpret the slope and y-intercept to derive insights.
• The slope informs how much the dependent variable (y) is expected to change with each unit change in the independent variable (x).
• A strong correlation coefficient indicates a reliable relationship. For instance, a high positive \(r\) values like 0.997 signifies that one can predict y with considerable accuracy using the regression line.

By combining both the regression line and correlation coefficient, you can glean a comprehensive understanding of the dataset's behavior. This enables you to predict future outcomes or trends effectively.