Chapter 10
Transforming Relationships
Day 2
AP Statistics
The Ladder of Power

Let’s examine the “0” rung of the ladder a
little more closely:
POWER
(x, ln(y))
Comment

This is very useful if the values of y increases by a
percentage.
(ln(x), y)

When x has a wide range or when the scatterplot
descends rapidly to the left and levels off to the
right
(ln(x), ln(y))

When one of the ladder powers is too big and the
other one is too small, this is often a useful
transformation. Also, if the scatterplot is
thickening, this transformation can be very useful.
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Non-Linear Regression

Let’s examine the relationship between shutter
speed and f/stops of a particular camera lens:
Shutter
speed
f/stop
Shutter
speed
f/stop
1/1000
2.8
1/60
11
1/500
4
1/30
16
1/250
5.6
1/15
22
1/125
8
1/8
32
3
Non-Linear Regression

Store the ln(F-stop) and ln(shutter speed) into L5
and L8, respectively:
POWER
(x, y)
=(speed, f/stop)
(x, ln(y))
=(speed, ln(f/stop))
(ln(x), y)
=(ln(speed), f/stop)
(ln(x), ln(y)) =(ln(speed), ln(f/stop))
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Non-Linear Regression

Let’s examine the relationship between shutter
speed and f/stops of a particular camera lens:
No!
Curved
No!
Curved
No!
Curved
YES!
Linear
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Non-Linear Regression

Let’s examine the relationship between shutter
speed and f/stops of a particular camera lens:

Although the data looks linear, it’s
still possible that it is actually
curved.
We need to check if this data is
actually linear or just appears to
be linear.

•
Let’s perform a residual plot on this
data.
6
Non-Linear Regression



Let’s examine the relationship between shutter
speed and f/stops of a particular camera lens:
The points appear to
have a random spread
about the MODIFIED
LSRL line. So, this seems
to be a good model to the
data - although it may
have increasing spread.
Be careful when
determining the actual
LSRL line.
7
Non-Linear Regression


Let’s examine the relationship between shutter
speed and f/stops of a particular camera lens:
The following is the modified equation:
ln( Fˆ  stop )  4 . 464  0 . 497 ln( shutter speed )


However, in the calculator we have the following:
yˆ  4 . 464  0 . 497 x
Take the first equation and solve for y-hat; this is the
true modified equation of the LSRL:
Fˆ  stop  e ( 4 .464  0 .497 ln( shutter
speed) )
or
Fˆ  stop  e^ ( 4 . 646  0 . 497 ln( shutter speed) )
8
Non-Linear Regression

Let’s examine the relationship between shutter
speed and f/stops of a particular camera lens:

Let’s make sure that our new equation fits the original
data:

Graph it:
It looks pretty good.
See if you can determine the
f/stop for a shutter speed of
¼ . Y1(1/4) = 43.612
9
Transformations
Type of Model Transformation
New Model
Re-expressed
Model
Exponential
( x , y )  ( x , ln( y ))
ln yˆ  a  bx
a  bx
yˆ  e
Logarithmic
( x , y )  (ln( x ), y )
yˆ  a  b ln( x )
yˆ  a  b ln( x )
Power
( x , y )  (ln( x ), ln( y ))
ln yˆ  a  b ln( x )
a  b ln( x )
yˆ  e
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Can’t We Just Use the Curve?
Although your calculator will do other types of
regression (quadratic, exponential, etc.), using
the curve has drawbacks.

•
•
First, lines are easy to understand. Using the curve,
throws out all of our understanding of linear
regression. We understand how to interpret the
slope and the y-intercept, and linear models are
more useful in advanced statistical practices. In
order to use the curve, we would have to come up
with a whole new system of understanding.
It’s best to use the linear model.
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Assignment
Chapter 10
Lesson:
Read:
Re-Expressing
Data
No new
reading
Problems:
Ch 10 Practice Quiz
Unit 2 Practice Test
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