There are several extensions of linear regression that apply to exponential
growth and power law models. Problems \(22-25\) will outline some of these
extensions. First of all, recall that a variable grows linearly over time if
it adds a fixed increment during each equal time period. Exponential growth
occurs when a variable is multiplied by a fixed number during each time
period. This means that exponential growth increases by a fixed multiple or
percentage of the previous amount. College algebra can be used to show that if
a variable grows exponentially, then its logarithm grows linearly. The
exponential growth model is \(y=\alpha \beta^{x}\), where \(\alpha\) and \(\beta\)
are fixed constants to be estimated from data.
How do we know when we are dealing with exponential growth, and how can we
estimate \(\alpha\) and \(\beta\) ? Please read on. Populations of living things
such as bacteria, locust, fish, panda bears, and so on tend to grow (or
decline) exponentially. However, these populations can be restricted by
outside limitations such as food, space, pollution, disease, hunting, and so
on. Suppose we have data pairs \((x, y)\) for which there is reason to believe
the scatter plot is not linear, but rather exponential, as described above.
This means the increase in \(y\) values begins rather slowly but then seems to
explode. Note: For exponential growth models, we assume all \(y>0\).
Consider the following data, where \(x=\) time in hours and \(y=\) number of
bacteria in a laboratory culture at the end of \(x\) hours.
$$
\begin{array}{l|rrrrr}
\hline x & 1 & 2 & 3 & 4 & 5 \\
\hline y & 3 & 12 & 22 & 51 & 145 \\
\hline
\end{array}
$$
(a) Look at the Excel graph of the scatter diagram of the \((x, y)\) data pairs.
Do you think a straight line will be a good fit to these data? Do the \(y\)
values seem almost to explode as time goes on?
(b) Now consider a transformation \(y^{\prime}=\log y .\) We are using common
logarithms of base 10 (however, natural logarithms of base \(e\) would work just
as well).
$$
\begin{array}{l|lllll}
\hline x & 1 & 2 & 3 & 4 & 5 \\
\hline y^{\prime}=\log y & 0.477 & 1.079 & 1.342 & 1.748 & 2.161 \\
\hline
\end{array}
$$
Look at the Excel graph of the scatter diagram of the \(\left(x,
y^{\prime}\right)\) data pairs and compare this diagram with the diagram in
part (a). Which graph appears to better fit a straight line?
(c) Use a calculator with regression keys to verify the linear regression
equation for the \((x, y)\) data pairs, \(\hat{y}=-50.3+32.3 x\), with correlation
coefficient \(r=0.882 .\)
(d) Use a calculator with regression keys to verify the linear regression
equation for the \(\left(x, y^{\prime}\right)\) data pairs,
\(y^{\prime}=0.150+0.404 x\), with correlation coefficient \(r=0.994\). The
correlation coefficient \(r=0.882\) for the \((x, y)\) pairs is not bad. But the
correlation coefficient \(r=0.994\) for the \(\left(x, y^{\prime}\right)\) pairs
is a lot better!
(e) The exponential growth model is \(y=\alpha \beta^{x}\). Let us use the
results of part (d) to estimate \(\alpha\) and \(\beta\) for this strain of
laboratory bacteria. The equation \(y^{\prime}=\) \(a+b x\) is the same as \(\log
y=a+b x .\) If we raise both sides of this equation to the power 10 and use
some college algebra, we get \(y=10^{a}\left(10^{b}\right)^{x}\). Thus \(\alpha
\approx 10^{a}\) and \(\beta \approx 10^{b}\). Use these results to approximate
\(\alpha\) and \(\beta\) and write the exponential growth equation for our strain
of bacteria.
Note: The TI-84Plus calculator fully supports the exponential growth model.
Place the original \(x\) data in list \(\mathrm{L} 1\) and the corresponding \(y\)
data in list \(\mathrm{L} 2 .\) Then press STAT, followed by CALC, and scroll
down to option 0: ExpReg. The output gives values for \(\alpha, \beta\), and the
correlation coefficient \(r\).
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