Question

Use a graphing calculator to find an equation for the line of best fit. $$ \begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 3 & 6 & 7 & 11 \\ \hline \boldsymbol{y} & 4 & 9 & 24 & 29 & 46 \\ \hline \end{array} $$

Short answer

Unfortunately, without using the actual calculator to input the values and compute the regression, it is not possible to provide a definitive short answer. However, the short answer will be in the form of \(y = ax + b\), where \(a\) and \(b\) are the slope and y-intercept values obtained from the graphing calculator.

Step-by-step solution

Data Input

Start by inputting each given point (x, y) into the graphing calculator. This is usually done by entering the x-values and y-values into separate lists.

Perform Linear Regression

Utilize the graphing calculator's function for linear regression. This is often accessed via the 'STAT' button, then by selecting 'CALC', and finally 'LinReg'. Make sure to specify the lists containing the x and y values.

Interpret Results

The calculator will provide an equation in the form \(y = ax + b\). The value \(a\) represents the slope of the line and \(b\) equates to the y-intercept. This is the equation for line of best fit.

Key concepts

Using a Graphing Calculator

A graphing calculator is an essential tool for solving algebraic and statistical problems, such as finding a line of best fit. To start, you need to input your data into the calculator. This typically means entering your x-values and y-values into separate lists. Each list should correspond to the specific axis on a graph.
Once your data is correctly entered, the calculator uses internal functions to process these values. It can compute various statistical calculations, such as linear regression. By using the graphing calculator, you can visualize the data points and their associated line of best fit. It helps to identify trends and patterns within the data, providing a visual representation that aids in understanding the relationship between variables.

• Ensure data is entered accurately.
• Familiarize yourself with the calculator functions to enhance efficiency.
• Use it to cross-check manual calculations or confirm results.

Understanding Linear Regression

Linear regression is a statistical method used to model and analyze the relationships between a dependent variable and one or more independent variables. When you perform linear regression on a graphing calculator, it will calculate the line of best fit through your data points.
The aim of linear regression is to find a linear equation that best describes the data. This line minimizes the sum of the squared differences between the observed values and the values predicted by the model.
The calculator does the heavy lifting by computing the necessary references from the data provided. This process is crucial for predicting outcomes and understanding the dynamics between different variables.

• Linear regression helps to find patterns within data.
• Essential for predictive modeling and analysis.
• Provides an equation helpful for further statistical evaluations.

Exploring Slope and Y-Intercept

The slope and y-intercept are key components of the linear equation used in linear regression. The general form of this equation is:\[ y = ax + b \]Here, \( a \) is the slope, and \( b \) is the y-intercept. In simple terms, the slope represents the rate at which the dependent variable changes with respect to the independent variable.
The y-intercept is where the line intersects the y-axis, indicating the value of \( y \) when \( x \) is zero. Understanding these components is crucial for interpreting the results provided by a graphing calculator.
By calculating the slope and y-intercept, you are also establishing a mathematical relationship that simplifies how changes in one variable might predict changes in another. This makes it easier to communicate and apply the data results in real-world scenarios.

• The slope indicates how steep the line is.
• The y-intercept shows where the line crosses the y-axis.
• Together, they provide a complete understanding of linear relationships.