Nonlinear Models and Scatter Plots

Contents: This page corresponds to § 4.6 (p. 372) of the text.

Suggested problems from the text:

p. 376 # 9 ,11, 13, 15, 19, 23, 27, 28, 29, 30


We have seen examples of fitting straight lines to scatter plots in the section Linear Models and Scatter Plots, but for some data sets no straight line is a good fit. In this section we will look at several different types of models that can be fit to data points.

Consider the following three scatter plots.

plot 1

plot 2

plot 3

Looking at the shapes of the plots, it appears that a straight line would fit plot 1 as well as any other shape, but plots 2 is shaped like an exponential curve, and plot 3 has the general shape of a logarithmic curve.

Example 1.

Make a scatter plot of the following ten points.

x

1

2

3

4

5

6

7

8

9

10

y

1.1

1.3

1.8

2.6

3.7

4.5

5.7

7.1

8.9

12.3

Scatter Plot for example 1

The next figure shows two models fit to the data points, one linear and one exponential.

The linear model is approximately y = 1.16*x - 1.45, with a correlation coefficient of 0.9572.

The exponential model is approximately y = 0.84*(1.31)^x, with a correlation coefficient of 0.9965.

Obviously the exponential curve provides a very good fit, and this is why its correlation coefficient is so close to 1.

Linear and Exponential

models fit to the data.

Exercise 1.

Use a graphing utility to plot the data points from example 1 and find the linear and exponential models that fit the data. Compare your answers to those given in the example.

If you are using a hand-held calculator, then section 5 of the appropriate chapter in the Graphing Technology Guide outlines the necessary steps.

The simplest way to do this problem is to use the Java Scatter Plot program. Using Scatter Plot involves the following steps.

  1. Enter the data points. For the first point you type 1,1.1 into the text box, then Enter. The second point is entered by typing 2,1.3 and enter, etc.
  2. After you have entered all ten points, draw the scatter plot by clicking the Plot button (top right).
  3. Compute the linear model by clicking the Linear button (below Plot). The correlation coefficient is displayed to the right of the Linear button, and the model is displayed below the button.
  4. Compute the exponential model by clicking the Exponential button (below Linear).

Exercise 2.

Make a scatter plot of the data points from example 3, p. 374 of the text. Fit a logarithmic model and a power model to the data points, and compare your results with the results of the example.