SECTION 12.2
TRANSFORMING TO
ACHIEVE LINEARITY
Transforming
• Nonlinear relationships between two quantitative variables
can sometimes be changed into linear relationships by
transforming one or both of the variables.
• Transformation is particularly effective when there is a
reason to think that the data are governed by some
nonlinear mathematical model.
• Once we transform the data to achieve linearity, we can
fit a least-squares regression line to the transformed
data and use this linear model to make predictions.
Power Models
• When theory or experience suggests that the relationship
between two variables follows a power model of the form
𝑦 = 𝑎𝑥 𝑝 , there are two options…
• Option 1: Raise the values of the explanatory variable x
to the power p, the look at a graph of (𝑥 𝑝 , 𝑦)
• Option 2: Take the pth root of the values of the response
variable y, then look at a graph of (𝑥, 𝑝 𝑦)
Example
• Imagine that you have been put in charge of organizing a
fishing tournament in which prizes will be given for the
heaviest fish caught. It would be easiest to measure the
length of the fish on the boat, but you need a way to
convert the length of the fish to its weight. You contact the
nearby marine research lab and they provide reference
data on several sizes of fish:
Transform power to linear
• Because length is one-dimensional and weight (like
volume) is three-dimensional, a power model of the form
𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑎(𝑙𝑒𝑛𝑔𝑡ℎ)3 should describe the relationship.
Option 1: (𝑙𝑒𝑛𝑔𝑡ℎ3, 𝑤𝑒𝑖𝑔ℎ𝑡)
• Write the equation of
the least-squares
regression line:
• Suppose someone
caught a fish that is 36
cm long. Predict their
weight:
• Interpret s:
3
Option 2: (𝑙𝑒𝑛𝑔𝑡ℎ, 𝑤𝑒𝑖𝑔ℎ𝑡)
• Write the equation
of the least-squares
regression line:
• Suppose someone
caught a fish that is
36 cm long. Predict
their weight:
• Interpret s:
Exponential and Log Models
• A useful strategy for straightening a curved pattern in a
scatterplot is to take the logarithm of one or both
variables.
• First – try to take the log or ln of y…if that doesn’t give you
a linear model…do both sides.
• If you log both – you are saying there is a power
relationship between the two variables.
Example
• Suppose you invest $100 in a savings account that pays
6% interest compounded annually. The table shows your
balance in the account after each of the first six years.
𝑦 = 100(1.06)𝑥
Example
One of the founders of Intel
Corporation predicted in 1965
that the number of transistors on
an integrated circuit chip would
double every 18 months.
•
When I take the ln(y)
Minitab output from transformed data
• Linear eq:
• Predict the number of transistors in 2020.
Example
• On July 31, 2005 a
team of astronauts
announced they
had discovered
what might be a
new planet. At the
time of discovery,
there were nine
planets in our solar
system. Here are
the data on the
distance from the
sun and period of
revolution of those
planets.
If I take the ln(y)
So I have to take the ln of both sides
Find the least-squares regression line
• Eq:
• Predict the period of revolution for Eris (the “new planet”),
which is 102.15 AU from the sun and show your work.
Homework
• Pg. 787 (35, 37-39, 42, 45-48)