In Exercise find the required linear model as follows: If you do not have a graphing calcu...

In Exercise find the required linear model as follows: If you do not have a graphing calculator or spreadsheet program, use the first and last data points to determine a line. If you do have a graphing calculator or spreadsheet program, find the least-squares regression line. (See Examples 1, 2, 4, and 6.)

Social Science The accompanying table shows how poverty-level income cutoffs (in dollars) for a family of four have changed over time (in large part because of inflation).†

(a) Find a linear− model for the data, with x = 0 corresponding to 1980.

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(b) Assuming the trend continues, what is the approximate poverty-level income cutoff in 2009? in 2011 ?

Example 6

Education Enrollment projections (in millions) for all U.S. colleges and universities in selected years are shown in the following table*:

(a) Let x = 0 correspond to 2000. Use a graphing calculator or spreadsheet program to find a linear model for the data and determine how well il fits the data points.

Solution The least-squares regression line (with coefficients rounded) is

 

as shown in Figure 1. The correlation coefficient is r ≈ .96, which is very close to 1, so this line fits the data well.

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(b) Assuming the trend continues, predict the enrollment in 2014.

Solution Let x = 14 (corresponding to 2014) in the regression equation:

 

Therefore, the enrollment in 2014 will be approximately 19,959,300 students.

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(c) According to this model, in what year will enrollment reach 21 million?

Solution Let y = 21 and solve the regression equation for x:

 

Since these enrollment figures change once a year, use the nearest integer value for x, namely, 17. So enrollment will reach 21 million in 2017.

FIGURE 1

*As of fall of each year; Statistical Abstracts of the United States: 2008.

†U.S. Census Bureau, Current Population Reports.