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. $$ (0,769),(1,677),(2,601),(3,543),(4,489),(5,411) $$

Short answer

The best-fitting model is the one with the highest correlation coefficient value, which indicates the best goodness of fit. The specific model (either linear or quadratic) will depend on the computed values for the given data set.

Step-by-step solution

Input Data

Enter the data points \((0,769),(1,677),(2,601),(3,543),(4,489),(5,411)\) into a graphing utility or a spreadsheet for analysis.

Linear Regression

Using the appropriate tool or function, generate a linear regression model for the entered data. This will produce a linear equation in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Quadratic Regression

Repeat the process, this time generating a quadratic regression model. The output will be a quadratic equation in the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

Model Selection

Compare the models using the goodness of fit - often indicated by the correlation coefficient, \(r\) or \(R^2\). The model with the highest value (closer to 1) is the best-fitting model for the data.