564 CHAPTER 10 Correlation and Regression Key Concept The preceding sections of this chapter deal with linear relationships only, but not all in the world is linear. This section is a brief introduction to methods for finding some nonlinear functions that fit sample data. We focus on the use of technology because the required calculations are quite complex. Shown below are five basic generic models considered in this section. Each of the five models is given with a generic formula along with an example of a specific function and its graph. 10-5 Nonlinear Regression Linear: y = a + bx Example: y = 1 + 2x Logarithmic: y = a + b In x Example: y = 1 + 2 In x Power: y = axb Example: y = 3x2.5 Exponential: y = abx Example: y = 2x Quadratic: y = ax2 + bx + c Example: y = x2 − 8x + 18 Here are three basic rules for identifying a good mathematical model: 1. Look for a pattern in the graph. Construct a graph, compare it to those shown here, and identify the model that appears to be most similar. 2. Compare values of R2. For each model being considered, use technology to find the value of the coefficient of determination R2. Choose functions that result in larger values of R2, because such larger values correspond to functions that better fit the observed sample data. ■ Don’t place much importance on small differences, such as the difference between R2 = 0.984 and R2 = 0.989. ■ Unlike in Section 10-4, we don’t need to use values of adjusted R2. Because the examples of this section all involve a single predictor variable, it makes sense to compare values of R2. ■ In addition to R2, another measure used to assess the quality of a model is the sum of squares of the residuals. See Exercise 18 “Sum of Squares Criterion.” 3. Think. Use common sense. Don’t use a model that leads to predicted values that are unrealistic. Use the model to calculate future values, past values, and values for missing data, and then determine whether the results are realistic and make sense. Don’t go too far beyond the scope of the available sample data.
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