Problem 23 - College Algebra and Calculus: An Applied Approach

Question

Use the regression capabilities of a graphing utility or a spreadsheet to find linear and quadratic models for the data. State which model best fits the data. $$ (-4,1),(-3,2),(-2,2),(-1,4),(0,6),(1,8),(2,9) $$

Step-by-Step Solution

Answer

When the regression is performed, it might result in a linear model and a quadratic model. The model (either linear or quadratic) with the lowest residuals is the one that best fits the data.

Step 1: Input Data Points

Import the given set of points into the spreadsheet or graphing tool.

Step 2: Perform Linear Regression

Generate a linear model for the data by performing a linear regression. In Excel, this can be done using the LINEST function. The linear model will be in the form \(y = mx + c\) where \(m\) is the slope and \(c\) is the y-intercept.

Step 3: Perform Quadratic Regression

Generate a quadratic model for the data by performing a quadratic regression. This can be done in Excel using the LINEST function on the y-values and the squares of the x-values. The quadratic model will be in the form \(y = ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants.

Step 4: Evaluate the Models

Examine the residuals (the difference between the real y values and the predicted y values) of both models. Pick the model with lowest residuals values as it is the one that best fits the data.

Key Concepts

Linear Regression

Linear regression is a statistical method that connects a set of data points with a straight line to model the relationship between two variables. It's typically used to determine if there's a linear association between the variables. This means one variable can help us predict the other.

A linear regression model takes the form of the equation \(y = mx + c\), where:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• \(m\) is the slope, indicating how much \(y\) changes for each unit change in \(x\).
• \(c\) is the y-intercept, the value of \(y\) when \(x = 0\).

To create a linear regression model, input your data into a tool like Excel, which can compute the best-fitting line using the LINEST function. This function determines the optimal \(m\) and \(c\) such that the sum of the squares of the residuals (differences between observed and predicted \(y\) values) is minimized. A smaller set of residuals indicates a better fit.

Quadratic Regression

Quadratic regression is a step beyond linear regression, where the model forms a parabola. This is particularly useful when data shows a trend that curves rather than follows a straight line. The quadratic model can capture more complex relationships.

In quadratic regression, the equation takes the form \(y = ax^2 + bx + c\), where:

• \(a\), \(b\), and \(c\) are constants that determine the parabola's shape.
• \(y\) changes with respect to \(x\) at a rate that accelerates or decelerates due to the \(ax^2\) term.

When conducting a quadratic regression using a tool like Excel, you can fit this model by using the LINEST function on your original \(y\) values and the squares of your \(x\) values. It effectively expands the capabilities of simple linear regression, allowing the equation to model curves, thereby often providing more accurate predictions especially when data forms a U-shape.

Model Fitting

Model fitting is the process of finding the best mathematical model that describes the relationship between variables in a set of data. It's essential for making valid predictions based on existing data.

The goal of model fitting is to ensure the chosen model makes accurate predictions about new or unseen data. This involves calculating the residuals, which are the differences between observed data points and the model's predicted values. The smaller the residuals, the better the fit.

When comparing linear and quadratic models, as in the exercise, evaluating the residuals of both models holds the key. You should select the model with the smallest residuals since it will likely predict more accurately. Furthermore, model complexity should align with the underlying data. A simpler model like linear regression is preferred if it provides a good fit as it reflects simplicity and generalization, while a quadratic model is more suited if there's significant curvature in the data.