4.3 Fitting linear models to data  (Page 4/14)

Using a regression line to make predictions

Gasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown in [link] . http://www.bts.gov/publications/national_transportation_statistics/2005/html/table_04_10.html Determine whether the trend is linear, and if so, find a model for the data. Use the model to predict the consumption in 2008.

Year	'94	'95	'96	'97	'98	'99	'00	'01	'02	'03	'04
Consumption (billions of gallons)	113	116	118	119	123	125	126	128	131	133	136

The scatter plot of the data, including the least squares regression line, is shown in [link] .

We can introduce new input variable, $t,$ representing years since 1994.

The least squares regression equation is:

$C(t)=113.318+2.209t$

Using technology, the correlation coefficient was calculated to be 0.9965, suggesting a very strong increasing linear trend.

Using this to predict consumption in 2008 $(t=14),$

$$\begin{array}{l}C(14)=113.318+2.209(14)\hfill \\ =144.244\hfill \end{array}$$

The model predicts 144.244 billion gallons of gasoline consumption in 2008.

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Key concepts

• Scatter plots show the relationship between two sets of data. See [link] .
• Scatter plots may represent linear or non-linear models.
• The line of best fit may be estimated or calculated, using a calculator or statistical software. See [link] .
• Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. See [link] .
• The correlation coefficient, $r,$ indicates the degree of linear relationship between data. See [link] .
• A regression line best fits the data. See [link] .
• The least squares regression line is found by minimizing the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables. See [link] .

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