AP Statistics
Notes
Name: ____________
Date: _____________
Lesson 4.1B: Modeling Exponential and Power Data
Learning Targets:
D: In the case of both exponential growth and power functions, perform a logarithmic
transformation and obtain points that lie in a linear pattern. Then use least-squares
regression on the transformed data. Carry out an inverse transformation to produce
a curve that models the original data.
E: Know that deviations from the overall pattern are most easily examined by fitting a
line to the transformed points and plotting the residuals from this line against the
explanatory variable (or fitted values).
How is the radius of a pizza related to the price?
The radius (in inches) and price (in dollars) of pizzas of different sizes from various pizzerias
are recorded below.
Radius
Price
(inches)
(dollars)
3
$2.50
4
$3.25
5
$5.00
6
$8.35
7
$11.95
8
$14.95
9
$19.75
10
$24.25
1. Place the radii in L1 and price in L2 and display a scatter plot of Price (y) against Radius (x).
a. Find the equation of the least squares regression line of Price against Radius
(PRICE = a + b(RADIUS)
b. Sketch the residual plot associated with the equation found in part a.
3
4
5
6
7
8
9
10
c. Is a linear model a good model for this data? Explain why or why not using the residual plot
in part b.
Steps for Modeling Non-Linear (Exponential or Power) Data
1.
2.
3.
4.
5.
Determine that the original data follows an exponential or power trend.
Use logarithms to transform the original data into a set of linear data.
Perform linear regression on the linearized data set.
Examine the residuals of the linearized data to assess the fit of the model.
Use properties of logarithms to transform the linear regression equation into an
exponential or power equation that fits the original data.
2. Transform the price to log(Price) and store them in L4. Plot the log(Price) against the radii
a. Find the equation of the least squares regression line of log(Price) against Radius
(log(PRICE) = a + b(RADIUS)
b. Transform the equation in part a. to an exponential function.
c. Use the calculator’s ExpReg feature to verify the equation found in part b.
d. Sketch the residual plot associated with the equation found in parts b and c.
3
4
5
6
7
8
9
10
e. Is an exponential model a good model for this data? Explain why or why not using the
residual plot in part d.
3. Transform the radius to log(Radius) and store them in L3.
Plot the log(Price) against the log(Radius)
a. Find the equation of the least squares regression line of log(Price) against log(Radius).
(log(PRICE) = a + b(log(RADIUS))
b. Transform the equation in part a. to a power function of the form y  a  x b
c. Use the calculator’s PowReg feature to verify the equation found in part b.
d. Sketch the residual plot associated with the equation found in parts b and c.
3
4
5
6
7
8
9
10
e. Is a power model a good model for this data? Explain why or why not using the residual
plot in part d.
4. Which model, the exponential or power, seems to be a better fit for the data?
Justify your answer
.