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Chapter 12 Section 2
Transforming to Achieve Linearity
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Transforming to Achieve Linearity
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Students will be able to:
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Use transformations involving powers and roots to achieve
linearity for a relationship between two variables.
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Use transformations involving logarithms to achieve linearity for a
relationship between two variables.
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Make predictions from a least-squares regression line involving
transformed data.
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Determine which of several transformations does a better job of
producing a linear relationship.
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Transforming to Achieve Linearity
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When two-variable data show a curved relationship, we must
develop new techniques for finding an appropriate model.
Once the data have been transformed to achieve linearity, we
can use least-squares regression to generate a useful model
for making predictions. If the conditions for regression
inference are met, we can estimate or test a claim about the
slope of the population regression line using the transformed
data.
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Transforming to Achieve Linearity
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Two types of transforming we will use: logarithm and square
root. When we apply a function such as one of these, this is
called transforming the data.
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Transforming data amounts to changing the scale of
measurement that was used when the data were collected.
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Linear transformations cannot straighten out curved
relationship between two variables. This is why we resort to
functions that are not linear.
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Transforming with Powers and
Roots
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Power model – dealing with circles, fish, people. We expect
area to go up with the square of a dimension such as
diameter or height. Volume should go up with the cube of a
linear dimension.
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Although a power model of the form y = axb describes the
relationship between x and y in each of these settings, there
is a linear relationship between xp and y.
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Take a look at the example on pp. 768-769 and 770
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Transforming to Achieve Linearity
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When experience or theory suggests that the relationship
between two variables is described by a power model of the
form y = axb, you now have two strategies for transforming
the data to achieve linearity
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Raise the values of the explanatory variable x to the p power and
plot the points (xp,y)
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Take the pth root for the values of the response variable y and
plot the points (x , square root p of y).
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If we do not know what power to choose then guess and test until
you find a transformation that works.
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Transforming with Logarithms
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Sometimes not all curved relationships are described by
power models but by a logarithm (base 10) – read paragraph
on pp. 771 under heading of logarithms for clarification.
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Exponential model: read paragraph on pp. 771
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Take a look at example on pp. 773-775
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Transforming with Logarithms
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Take a look at the rest of the examples for clarification
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I would work the odd problems in the practice section for
further understanding.