AP Statistics Section 4.1 B
Power Law Models
Exponential growth occurs when a
variable is multiplied by a fixed
number in each equal time period.
In other words, exponential
growth increases by a fixed percent
of the previous total in each equal
time period.
Example 2: You won a lottery. You
have a choice of (a) getting
$100,000 each day for the month
of November or (b) getting $0.01
on November 1st, $0.02 on
November 2nd, $0.04 on November
3rd, etc until November 30th. What
is your choice?
For choice (a), at the end of the
month you would have
30  100000  $ 3 , 000 , 000
For choice (b), you would have
p (1  r )
t
. 01 (1  1)
30
$ 10 , 737 , 418 . 24
This example illustrates the power
of exponential growth.
Example 3: Here is the number of bacteria present
after a given number of hours.
Graph the data.
The curve is increasing exponentially. How can
we tell if an exponential model is appropriate?
For equal increments in x, calculate the ratio
of a y-value divided by the previous y-value.
All such ratios should be equal for a perfectly
exponential function.
2 .4
1 .8
 1. 3
3.1
2.4
 1 . 29
4.3
3.1
 1 . 39
Exponential growth can be
modeled by the equation
x
________,
y  a  b where a and b are
constants.
To help us “straighten” this model
we will use a logarithm
transformation.
Take the ln of both sides.
ln y  ln( a  b )
x
The ln of a product equals the sum of the ln’s
.
ln y  ln a  ln b )
x
The ln of a power equals the power times the ln.
ln y  ln a  x ln b )
Notice that this form is the equation of a line.
For a logarithmic transformation
we will take the log of the y-values
and leave the x-values alone.
Graph hours (x) in L1 vs. ln y in L3.
Calculate the LSL on the transformed points and
construct a residual plot to verify the validity of our
model.
ln(# bacteria )   . 0047  . 5860 (# hours )
Determine r2 and interpret it in the context of
the problem.
r  . 999
2
99.9% of the variation in the natural log of
the number of bacteria is accounted for by
the linear relationship with the number of
hours
Interpret the slope and y-intercept in the
context of the problem.
The slope tells us that an increase in one hour
will cause an increase of .5860 in the natural log
of the number of bacteria.
The y-intercept tell us that at hour 0, the
natural log of the number of bacteria will be
-.0047
Solve the LSL for yˆ and predict how many
bacteria will be present at 3.75 hours.
ln yˆ   . 0047  . 5860 x
yˆ  e
yˆ  e
 . 0047  . 5860 x
 . 0047
yˆ  8 . 96
e
. 5860 x
Example 4: In 1965, Gordon Moore, a founder of
Intel Corp., predicted that the number of
transistors in an integrated circuit chip would
double every 18 months (i.e exponential
growth). Here is some data from a stats package
that was run on the data of the number of
transistors versus time since 1970. The data was
straightened out by using a logarithmic
transformation on the number of transistors. x =
# of years since 1970, y = ln (transistors). We use
“years since” so that the values are smaller and
don’t create overflow problems with the
calculator.
Predictor
Coef
Constant
7.4078
Yrs since1970 0.3316
R-Sq = 99.5%
Write the LSL.
ln(# transistor s)  7 . 4078  . 3316 ( years since 1970)
2
Interpret r in context.
99.5% of the variation in ln(#transistors) is
accounted for by the linear relationship
with the # of years since 1970.
Predict the number of transistors in
the year 2003.
yˆ  e
7 . 4078  . 3316 ( 33 )
yˆ  93 , 231 ,853 . 78