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Part II: Making Predictions by Using Regression
hours when using this model because the scores on this quiz don’t go above
ten. The model likely levels off after six hours to a score of ten, indicating
that studying more than six hours is overkill.
Going Up? Going Down? Go Exponential!
Exponential models work well in situations where a y variable either increases
or decreases exponentially over time. That means, the y variable starts out
slow, then increases at a faster and faster rate, or it starts out high and
decreases at a faster and faster rate. Many processes in the real world
behave like an exponential model: for example, population size over time,
average household incomes over time, the length of time a product lasts, or
the level of patience one has as the number of statistics homework problems
goes up (of course, using this book should cut that time in half, no?).
In this section, you familiarize yourself with the exponential regression
model, and see how to use it to fit data that either rises or falls at an expo-
nential rate. You also discover how to build and assess exponential regres-
sion models to make accurate predictions for a response variable y, using an
explanatory variable x.
Recollecting exponential models
x
Exponential models have the form y = αβ . These models involve a constant,
β, raised to higher and higher powers of x multiplied by a constant, α. The
constant β represents the amount of curvature in the model. The constant α
is a multiplier in front of the model that shows where the model crosses the
y-axis (because when x = 0, y = α * 1).
An exponential model generally looks like the upper part of a hyperbola
(remember those from advanced algebra?). A hyperbola is a curve that
crosses the y-axis at a point and curves downward toward zero or starts
at some point and curves upward to infinity (see Figures 7-8a and 7-8b for
examples). If β is greater than one in an exponential model, the graph curves
upward toward infinity. If β is less than one, the graph curves downward
toward zero. All exponential models stay above the x-axis.
x
For example, the model y = 1 3 is an exponential model. Here, say you
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made α = 1, indicating that the model crosses the y-axis at 1 (because plug-
ging x = 0 into the equation gives you 1). You set the value of β equal to three,