2.4 Fitting Linear Models to Data
In the real world, rarely do things follow trends perfectly. When we expect the trend to behave linearly, or when inspection suggests the trend is behaving linearly, it is often desirable to find an equation to approximate the data. Finding an equation to approximate the data helps us understand the behavior of the data and allows us to use the linear model to make predictions about the data, inside and outside of the data range.
Example 1
The table below shows the number of cricket chirps in 15 seconds, and the air temperature, in degrees Fahrenheit[1]. Plot this data, and determine whether the data appears to be linearly related.
Plotting this data, it appears there may be a trend, and that the trend appears roughly linear, though certainly not perfectly so.
The simplest way to find an equation to approximate this data is to try to “eyeball” a line that seems to fit the data pretty well, then find an equation for that line based on the slope and intercept.
You can see from the trend in the data that the number of chirps increases as the temperature increases. As you consider a function for this data you should know that you are looking at an increasing function or a function with a positive slope.
Flashback
 1. a. What descriptive variables would you choose to represent Temperature & Chirps?
 b. Which variable is the independent variable and which is the dependent variable?
 c. Based on this data and the graph, what is a reasonable domain & range?
 d. Based on the data alone, is this function onetoone, explain?
Example 2
Using the table of values from the previous example, find a linear function that fits the data by “eyeballing” a line that seems to fit.
On a graph, we could try sketching in a line. The scale on the axes has been adjusted to including the vertical axis in the graph.
Using the starting and ending points of our “hand drawn” line, points (0, 30) and (50, 90), this graph has a slope of and a vertical intercept at 30, giving an equation of
where c is the number of chirps in 15 seconds, and T(c) is the temperature in degrees Fahrenheit.
This linear equation can then be used to approximate the solution to various questions we might ask about the trend. While the data does not perfectly fall on the linear equation, the equation is our best guess as to how the relationship will behave outside of the values we have data for. There is a difference, though, between making predictions inside the domain and range of values we have data for, and outside that domain and range.
Interpolation and Extrapolation
Interpolation: When we predict a value inside the domain and range of the data
Extrapolation: When we predict a value outside the domain and range of the data
For the Temperature as a function of chirps in our hand drawn model above:
Interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44.
Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or greater than 44.
Example 3
a) Would predicting the temperature when crickets are chirping 30 times in 15 seconds be interpolation or extrapolation? Make the prediction, and discuss if it is reasonable.
b) Would predicting the number of chirps crickets will make at 40 degrees be interpolation or extrapolation? Make the prediction, and discuss if it is reasonable.
With our cricket data, our number of chirps in the data provided varied from 18.5 to 44. A prediction at 30 chirps per 15 seconds is inside the domain of our data, so would be interpolation. Using our model:
degrees.
Based on the data we have, this value seems reasonable.
The temperature values varied from 52 to 80.5. Predicting the number of chirps at 40 degrees is extrapolation since 40 is outside the range of our data. Using our model:
Our model predicts the crickets would chirp 8.33 times in 15 seconds. While this might
be possible, we have no reason to believe our model is valid outside the domain and range. In fact, generally crickets stop chirping altogether below around 50 degrees.
When our model no longer applies after some point, it is sometimes called model breakdown.
Try it Now
What temperature would you predict if you counted 20 chirps in 15 seconds?
Fitting Lines with Technology
While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data values[2]. This technique is called leastsquare regression, and can be computed by many graphing calculators, spreadsheet software like Excel or Google Docs, statistical software, and many webbased calculators[3].
Example 4
Find the leastsquares regression line using the cricket chirp data from above.
Using the cricket chirp data from earlier, with technology we obtain the equation:
Notice that this line is quite similar to the equation we “eyeballed”, but should fit the data better. Notice also that using this equation would change our prediction for the temperature when hearing 30 chirps in 15 seconds from 66 degrees to:
degrees.
Most calculators and computer software will also provide you with the correlation coefficient, a measure of how closely the line fits the data.
Correlation Coefficient
The correlation coefficient is a value, r, between 1 and 1.
r > 0 suggests a positive (increasing) relationship
r < 0 suggests a negative (decreasing) relationship
The closer the value is to 0, the more scattered the data
The closer the value is to 1 or 1, the less scattered the data is
The correlation coefficient provides an easy way to get some idea of how close to a line the data falls.
We should only compute the correlation coefficient for data that follows a linear pattern; if the data exhibits a nonlinear pattern, the correlation coefficient is meaningless. To get a sense for the relationship between the value of r and the graph of the data, here are some large data sets with their correlation coefficients:
Examples of Correlation Coefficient Values
Example 5
Calculate the correlation coefficient for our cricket data.
Because the data appears to follow a linear pattern, we can use technology to calculate r = 0.9509. Since this value is very close to 1, it suggests a strong increasing linear relationship.
Example 6
Gasoline consumption in the US has been increasing steadily. Consumption data from 1994 to 2004 is shown below.[5] Determine if the trend is linear, and if so, find a model for the data. Use the model to predict the consumption in 2008.
To make things simpler, a new input variable is introduced, t, representing years since 1994.
Using technology, the correlation coefficient was calculated to be 0.9965, suggesting a very strong increasing linear trend.
The leastsquares regression equation is:
.
Using this to predict consumption in 2008 (t = 14),
billions of gallons
The model predicts 144.244 billion gallons of gasoline will be consumed in 2008.
Try it Now
2. Use the model created by technology in example 6 to predict the gas consumption in 2011. Is this an interpolation or an extrapolation?
Important Topics of this Section
Fitting linear models to data by hand
Fitting linear models to data using technology
Interpolation
Extrapolation
Correlation coefficient
Flashback Answers
1. a. T = Temperature, C = Chirps (answers may vary)
b. Independent (Chirps) , Dependent (Temperature)
c. Reasonable Domain (18.5, 44) , Reasonable Range (52, 80.5) (answers may vary)
d. NO, it is not onetoone, there are two different output values for 35 chirps.
Try it Now Answers
1. 54 degrees Fahrenheit
2. 150.871 billion gallons, extrapolation
Section 2.4 Exercises
 The following is data for the first and second quiz scores for 8 students in a class. Plot the points, then sketch a line that fits the data.
First Quiz  11  20  24  25  33  42  46  49 
Second Quiz  10  16  23  28  30  39  40  49 
 Eight students were asked to estimate their score on a 10 point quiz. Their estimated and actual scores are given. Plot the points, then sketch a line that fits the data.
Predicted  5  7  6  8  10  9  10  7 
Actual  6  6  7  8  9  9  10  6 
Based on each set of data given, calculate the regression line using your calculator or other technology tool, and determine the correlation coefficient.
3. 
 4. 
 5. 
 6. 

 A regression was run to determine if there is a relationship between hours of TV watched per day (x) and number of situps a person can do (y). The results of the regression are given below. Use this to predict the number of situps a person who watches 11 hours of TV can do.
y=ax+b
a=1.341
b=32.234
r2=0.803
r=0.896
 A regression was run to determine if there is a relationship between the diameter of a tree (x, in inches) and the tree’s age (y, in years). The results of the regression are given below. Use this to predict the age of a tree with diameter 10 inches.
y=ax+b
a=6.301
b=1.044
r2=0.940
r=0.970
Match each scatterplot shown below with one of the four specified correlations.
9. r = 0.95 10. r = 0.89 11. r = 0.26 12. r = 0.39
A B C D
 The US census tracks the percentage of persons 25 years or older who are college graduates. That data for several years is given below. Determine if the trend appears linear. If so and the trend continues, in what year will the percentage exceed 35%?
Year  1990  1992  1994  1996  1998  2000  2002  2004  2006  2008 
Percent Graduates  21.3  21.4  22.2  23.6  24.4  25.6  26.7  27.7  28  29.4 
 The US import of wine (in hectoliters) for several years is given below. Determine if the trend appears linear. If so and the trend continues, in what year will imports exceed 12,000 hectoliters?
Year  1992  1994  1996  1998  2000  2002  2004  2006  2008  2009 
Imports  2665  2688  3565  4129  4584  5655  6549  7950  8487  9462 
[1] Selected data from http://classic.globe.gov/fsl/scientistsblog/2007/10/. Retrieved Aug 3, 2010
[2] Technically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values.
[3] For example, http://www.shodor.org/unchem/math/lls/leastsq.html
Contributors
 David Lippman (Pierce College)
 Melonie Rasmussen (Pierce College)