In the following problem, determine whether a straight line is a good model for the data. You may do this either visually, by plotting the data points, or analytically, by finding the correlation coefficient for the least-squares regression line. (See Examples 5 and 6.)
Natural Science The accompanying table shows the average monthly temperature (in degrees Fahrenheit) in Cleveland, Ohio, based on data from 1961 to 1990.‡ Let x = 2 correspond to February, x = 4 to April, etc.
Month | Feb | April | June | Aug | Oct | Dec |
Temperature | 27.3 | 47.5 | 67.5 | 70.3 | 52.7 | 30.9 |
Example 6
Education Enrollment projections (in millions) for all U.S. colleges and universities in selected years are shown in the following table*:
Year | Enrollment | Year | Enrollment |
2000 | 15.313 | 2005 | 16.679 |
2001 | 15.928 | 2006 | 16.887 |
2002 | 16.103 | 2007 | 17.958 |
2003 | 16.360 | 2008 | 18.264 |
2004 | 16.468 |
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(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.
(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.
(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
‡National Climatic Research Center.
*As of fall of each year; Statistical Abstracts of the United States: 2008.
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