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.)

Physical Science While shopping for an air conditioner, Mario Cekada consulted the following table giving a machine’s BTUs and the square footage (ft2) that it would cool‡:

(a) Find a linear model for the data.

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(b) To check the fit of the data to the line, use the results from part (a) to find the number of BTUs required to cool rooms of 150 ft2, 280 ft2, and 420 ft2. How well do the actual data agree with the predicted values?

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(c) Suppose Adam’s room measures 235 ft2. Use the results from part (a) to decide how many BTUs it requires. If air conditioners are available only with the BTU choices in the table, which should Adam choose?

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.

‡Morris Carey and James Carey, “On the House,” Sacramento Bee, July 29, 2000.