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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] .

Scatter plot, showing the line of best fit. It is titled 'Gas Consumption VS Year'. The x-axis is 'Year After 1994', and the y-axis is 'Gas Consumption (billions of gallons)'. The points are strongly positively correlated and the line of best fit goes through most of the points completely.

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

The least squares regression equation is:

C ( t ) = 113.318 + 2.209 t

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 ) ,

C ( 14 ) = 113.318 + 2.209 ( 14 )           = 144.244

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

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Use the model we created using technology in [link] to predict the gas consumption in 2011. Is this an interpolation or an extrapolation?

150.871 billion gallons; extrapolation

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Access these online resources for additional instruction and practice with fitting linear models to data.

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] .

Section exercises

Verbal

Describe what it means if there is a model breakdown when using a linear model.

When our model no longer applies, after some value in the domain, the model itself doesn’t hold.

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What is interpolation when using a linear model?

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What is extrapolation when using a linear model?

We predict a value outside the domain and range of the data.

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Explain the difference between a positive and a negative correlation coefficient.

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Explain how to interpret the absolute value of a correlation coefficient.

The closer the number is to 1, the less scattered the data, the closer the number is to 0, the more scattered the data.

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Algebraic

A regression was run to determine whether there is a relationship between hours of TV watched per day ( x ) and number of sit-ups a person can do ( y ) . The results of the regression are given below. Use this to predict the number of sit-ups a person who watches 11 hours of TV can do.

y = a x + b a = −1.341 b = 32.234   r = −0.896
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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