1.2.1 Finding Models

When a relationship between varying quantities is suggested by a scatter plot, we may want to describe it mathematically by an equation that summarizes the way the two variables are related. Such an equation is called a mathematical model.

A good model simplifies the phenomenon it represents and gives us the ability to predict. If we can find an equation of a line or curve that closely fits a scatter plot, we can focus on the important characteristics of the relationship between the variables without the clutter of a scatter plot. We can also use this equation to predict the values of one variable for specific values of the other variable. Sometimes we use the model to interpolate, or estimate values between observed values. Sometimes we use the model to extrapolate, or predict values outside the region of observations.

Table 1 shows estimates for the per capita consumption of cigarettes by adults (18 years old and older) in the United States in the period between 1975 and 2000. The data are plotted in Figure 1.

Note 1 – Data source

Activity 1

1. Use your graphing tool to plot the data in Table 1. (Click on the appropriate computer algebra icon at the right, or use your graphing calculator.)

2. Find an equation for your line of the form $y=mt+b$ where t stands for the year and y stands for per capita cigarette consumption.

3. Use your model to estimate per capita cigarette consumption in the years 1985 and 1990 and compare to the data in the table.

4. Now use your model to estimate the per capita cigarette consumption in 2002.

Comment on Activity 1

Have you noticed that very few of our activities and exercises have clearly determined "right" answers? That's often the way mathematics works in the real world. There are answers that are more sensible and answers that are less sensible and answers that are clearly wrong — but not always answers that are clearly right. Your job is to find answers that are sensible — and to be prepared to defend those answers.

Activity 2

Table 2 and Figure 2 show data on the number of daily newspapers in the United States between 1990 and 2002. The graph shows a clear decreasing trend. Use your graphing tool to find a straight line that approximates this data. (Click on a computer algebra icon at the right, or use your graphing calculator.) How can you check your equation?

Number of Daily Newspapers in US            

Table 2	Figure 2
Year Number of Papers 1990 1,611 1991 1,586 1992 1,570 1993 1,556 1994 1,548 1995 1,533 1996 1,520 1997 1,509 1998 1,489 1999 1,483 2000 1,480 2001 1,468 2002 1,457

Note 2 – Data source

Comment on Activity 2

Checkpoint 1

Contents for Chapter 1