AP STATISTICS
Section 4.1 Transforming to Achieve Linearity
Objective: To be able to use least squares regression line
techniques to develop a model for non-linear data.
Goals:
1. When working with non-linear data we want to apply a
function to one or both variables in order to “straighten”
the data.
2. Next, we will use LSRL techniques to develop a model
for the “transformed” data.
3. Last, use the inverse function to develop a model for the
original relationship.
Review logarithms: (Think exponents)
1. Definition: 𝑙𝑜𝑔𝑏 𝑥 = 𝑎 if and only if 𝑏 𝑎 = 𝑥
Ex.
2. 𝑙𝑜𝑔 𝐴𝐵 = log 𝐴 + log 𝐵
3. 𝑙𝑜𝑔 𝐴/𝐵 = log 𝐴 − log 𝐵
4. 𝑙𝑜𝑔 𝐴𝑝 = 𝑝 log 𝐴
5. Inverse function:
10𝑙𝑜𝑔10 𝑏 = 𝑏
Base 10: log10 𝑥 or log x
Base e: 𝑙𝑛𝑥
Linear growth: (𝑦 = 𝑎 + 𝑏𝑥) As x increases in fixed
increments, we add a constant to the y values.
Exponential growth: 𝑦 = 𝑎𝑏 𝑥 As x increases in fixed
increments, we multiply the y-values by a constant.
If b > 1 then we have exponential growth.
If 0 < b < 1 then we have exponential decay.
Outline for Creating a Model for Exponential Data:
𝑦 = 𝑎𝑏 𝑥
1. Plot the data
2. If you think it may be linear, try linear regression
methods and create a residual plot. If it is not linear,
you will see a curve in the residual plot.
3. Examine common ratios. 𝐶𝑅 =
𝑦𝑛
.
𝑦𝑛−1
We can only
compare CRs over fixed increments of x.
If exponential, then all the CR’s should be approximately
the same. Calculate the average common ratio.
4. Calculate log 𝑦
5. Plot log 𝑦 vs. x. If this is a successful transformation,
then the scatterplot should appear straight.
6. Use LSRL techniques to develop a model for log 𝑦 =
𝑎 + 𝑏𝑥
7. Check the residual plot for the transformed model.
8. Use the inverse function to develop a model for the
original data.
Ex. Create a model for the NCAA March Madness
Tournament that plots the number of teams versus the
round of the tournament.
Data set:
Outline for Creating a Model for Power Data: 𝑦 = 𝑎𝑥 𝑏
1. Plot the data.
2. Calculate log 𝑥 and log 𝑦
3. Plot log 𝑦 vs. log 𝑥 . If this is a successful
transformation, then the scatterplot should appear
straight.
4. Use LSRL techniques to develop a model for
log 𝑦 = 𝑎 + 𝑏 log 𝑥
5. Check the residual plot for the transformed model.
6. Use the inverse function to develop a model for the
original data.
• When checking the final model on your calculator be sure
to change x to log 𝑥 to avoid any overflow errors.
• A good indicator that a power model may be better than
an exponential is that the x values cover a large range of
values.
Ex. Planetary data. Use a planet’s average distance from
the sun to create a model to predict it’s year.
A.U.: 0.39, 0.72, 1.00, 1.52, 5.20, 9.54, 19.19, 30.06,
39.53
Year: 0.24, 0.61, 1.00, 1.88, 11.86, 29.46, 84.07, 164.82,
247.68
Work:
Ex. Use the number of days alive to predict the body
weight of Mr. Reid’s dog.
Days:
1,
51,
64,
85,
118
Weight: 0.625, 9.900, 12.750, 19.000, 35.700