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Module 3 – Exponential Regression
Exponential Regression
This module of study presents and illustrates regression analysis for exponential
functions. This module of study builds upon the previous module of study of regression
analysis for linear functions. In this module, we will study an example of exponential
growth and an example of exponential decay.
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Module 3 – Exponential Regression
Cellular Telephone Subscribers
We begin with an example of the use of regression analysis for exponential growth.
This example makes use of the following data from the U. S. Census Bureau on the
number of cellular telephone subscribers in the United States:
Calendar
Year
1990
1995
1996
1997
1998
1999
2000
2001
Year
X
0
5
6
7
8
9
10
11
Actual Cellular
Telephone
Subscribers
(millions)
Y
5.3
33.8
44.0
55.3
69.2
86.0
109.5
128.4
When cellular telephones first became available, they were unproven, somewhat
unreliable and expensive. As the technology was improved and advanced with new
features such as text messaging and digital pictures, and as the cost and reliability
became more practical, usage began to accelerate ever more rapidly from year to year.
This phenomenon represents an example of exponential growth.
After entering the x-data into list L1 and the y-data into list L2 in our calculator via the
STAT EDIT function, we can use the ExpReg function from the STAT CALC menu to
obtain an exponential regression equation that serves as the model of this exponential
growth:
ExpReg
Y a 
bX
a 6.622854813
b 1.333170131
r 2 0.9755769817
r 0.9877130057
The correlation coefficient r 0.9877130057 confirms our expectation that this data is
exponential in nature.
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Module 3 – Exponential Regression
If we truncate the digits of precision in the numbers that were calculated above by the
ExpReg function, the exponential equation that we will use for this example is:
Y 6.6 
(1.33) X
This exponential regression equation is the model of the growth of cellular telephone
subscribers per year in the given calendar period. If we enter this exponential
regression equation into our calculator, we can obtain the following table of values:
Year
Predicted
Cellular
Telephone
Subscribers
(millions)
Y
6.6
27.5
36.5
48.6
64.6
85.9
114.3
152.0
X
0
5
6
7
8
9
10
11
Cellular Subscribers (Millions)
Next, we can draw a scatter plot diagram and we can superimpose the graph of the
exponential regression equation on the scatter plot diagram:
160
140
120
100
80
60
40
20
0
0
5
10
Year
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Module 3 – Exponential Regression
Next, we calculate the standard error of the estimate from the following data:
Year
X
0
5
6
7
8
9
10
11
Actual
Predicted
Cellular
Cellular
Telephone
Telephone
Subscribers Subscribers
(millions)
(millions)
Y
Y
5.3
6.6
33.8
27.5
44.0
36.5
55.3
48.6
69.2
64.6
86.0
85.9
109.5
114.3
128.4
152.0
Residual
Square of
Residual
-1.3
6.3
7.5
6.7
4.6
0.1
-4.8
-23.6
1.69
39.69
56.25
44.89
21.16
0.01
23.04
556.96
The residual is the difference between the actual number and the predicted number of
cellular telephone subscribers. For any given year, it is the measure of the vertical gap
between the actual number of subscribers on the scatter plot diagram and the predicted
number of subscribers on the regression curve.
Next, we obtain the average value of the eight numbers in the last column of this table,
that is, the sum of these numbers divided by eight:
743 .69
92.96125
8
The standard error of the estimate is the square root of this average:
Standard Error of the Estimate =
92.96125 9.6
Then, we use the regression equation to predict the number of subscribers in the
succeeding year on the assumption that cellular telephone growth continues at an
accelerated pace. In calendar year 2002, that is, for year X = 12, the predicted number
of subscribers would be:
Y 6.6 
(1.33) 202.2
12
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Module 3 – Exponential Regression
To calculate a confidence interval around this prediction, we multiply the standard error
of the estimate by 2:
2 9.6 19.2
The confidence interval for the predicted number of subscribers is as follows. For 2002,
there is a 95% likelihood that the number of subscribers is in the following interval:
202.2 – 19.2 = 183.0
202.2
202.2 + 19.2 = 221.4
Predicted Value Minus
Two Times the
Standard Error of the
Estimate
Predicted
Value
Predicted Value Plus
Two Times the
Standard Error of the
Estimate
The data for cellular telephone growth in this example was obtained from the U. S.
Census Bureau and this example was published in the following textbook:
“Applying Algebraic Thinking to Data,” by Phil DeMarois, Mercedes McGowen,
Darlene Whitkanack, Third Edition, Copyright 2006, ISBN 0-7575-2918-6,
Kendall/Hunt Publishing Company, pp. 260-265.
Curve of Technology Adoption
Predictions of the future are perilous. In this study of cellular telephone use, we cannot
expect that the model of exponential growth for cellular telephone subscribers would be
sustained indefinitely. In fact, if we examine the table of residuals, we observe that the
size of the negative residual value of -23.6 for year 11 already indicates that the actual
number of cellular telephone subscribers is beginning to lag behind the number of
cellular telephone subscribers that is predicted by the exponential growth model.
This is an expected happening in the world of technology. As a new technology is
discovered and launched, intrepid and eager adopters acquire and use that technology.
However, these early adopters are in the minority. As the new technology evolves and
becomes more widely accepted, mainstream acceptance ensues in an accelerated
manner for an extended period of time.
Finally, as the technology in question matures and is overtaken downstream by other
discoveries and competitive technologies, the market for that technology stops growing
rapidly and settles eventually onto a plateau. The manner in which a new technology is
received by the marketplace is the subject of the following book:
“Crossing the Chasm,” by Geoffrey A. Moore, Copyright 1991, ISBN 0-06662-002-3,
Harper Collins Publishers.
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Module 3 – Exponential Regression
Cellular Subscribers (Millions)
The following graph is a conjectured projection of how we might expect the market for
cellular telephones to unfold:
300
250
200
150
100
50
0
0
5
Early
Adopters
10
Year
15
Mainstream
Acceptance
20
Mature
Market
This is an example of what is known as the logistical model of growth. We will examine
this model in a subsequent module of study and we will discover that this model is an
excellent fit for the pattern that is present in the cellular telephone subscriber data.
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Module 3 – Exponential Regression
Concentration of Drug in Bloodstream
In this example of exponential regression, we will use an exponential function to
measure the decline in the amount of a drug in the bloodstream of a hospital patient in
the hours after the drug is injected. This will be an example of exponential decay.
Equations such as this come about by conducting experiments, making careful
measurements and performing analysis of the measured data. Assume that 600
milligrams of a drug are injected into the bloodstream of a patient as an experiment.
Then, the amount of the drug remaining in the bloodstream is measured over a period
of hours:
Hour
X
0
1
2
3
4
5
Measured
Milligrams of
the Drug in the
Bloodstream
Y
596
326
129
87
34
23
After entering the x-data into list L1 and the y-data into list L2 in our calculator via the
STAT EDIT function, we can use the ExpReg function from the STAT CALC menu to
obtain an exponential regression equation that serves as the model of this exponential
decline:
ExpReg
Y a 
bX
a 583.5120528
b 0.5117195853
r 2 0.9888898691
r 0.9944294189
The correlation coefficient r 0.9944294189 confirms our expectation that this data is
exponential in nature. Note that this correlation coefficient is negative which indicates
that this is an example of exponential decay rather than exponential growth.
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Module 3 – Exponential Regression
If we truncate the digits of precision in the numbers that were calculated above by the
ExpReg function, the exponential equation that we will use for this example is:
Y 583 
(0.51) X
If we enter this exponential regression equation into our calculator, we obtain the
following table of values:
Hour
Predicted
Milligrams of
the Drug in the
Bloodstream
Y
583
297
152
77
39
20
X
0
1
2
3
4
5
Next, we can draw a scatter plot diagram and we can superimpose the graph of the
exponential regression equation on the scatter plot diagram:
600
Milligrams of Drug
500
400
300
200
100
0
0
1
2
3
Hours
8
4
5
6
Module 3 – Exponential Regression
Next, we calculate the standard error of the estimate from the following data:
Hour
X
0
1
2
3
4
5
Measured
Milligrams of
the Drug in
the
Bloodstream
Y
596
326
129
87
34
23
Predicted
Milligrams of
the Drug in
the
Bloodstream
Y
583
297
152
77
39
20
Residual
Square of
Residual
13
29
-23
10
-5
3
169
841
529
100
25
9
The residual is the difference between the measured number and the predicted number
of milligrams. For any given hour, it is the measure of the vertical gap between the
measured number of milligrams on the scatter plot diagram and the predicted number of
milligrams on the regression curve.
Next, we obtain the average value of the six numbers in the last column of this table,
that is, the sum of these numbers divided by six:
1673
278.83
6
The standard error of the estimate is the square root of this average:
Standard Error of the Estimate =
278.83 16.7
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Module 3 – Exponential Regression
Then, we could use the regression equation as a basis for predicting the number of
milligrams of this drug that should be expected to be remaining in the bloodstream of a
future patient at a given time, for example, X = 4 hours after being injected:
Y 583 
(0.51) 4 39
To calculate a confidence interval around this prediction, we multiply the standard error
of the estimate by 2:
2 16.7 33.4
The confidence interval for the predicted number of milligrams is determined as follows.
At X = 4 hours, there is a 95% likelihood that the number of milligrams that would be
expected to be in the bloodstream of a given patient is in the following interval:
39 – 33.4 = 5.6
39
39 + 33.4 = 72.4
Predicted Value Minus
Two Times the
Standard Error of the
Estimate
Predicted
Value
Predicted Value Plus
Two Times the
Standard Error of the
Estimate
This type of analysis allows the medical staff to know how long to wait before giving a
patient a subsequent injection.
In a similar manner, an exponential regression equation could be used to analyze and
predict the remaining blood alcohol content over a period of time after a person drinks a
given amount of an alcoholic beverage.
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