CHAPTER 4: MORE ON TWO-VARIABLE DATAPart 2

POWER LAW MODELS

Recall that exponential growth models become linear when we apply the logarithm transformation to the response variable y.

Similarly, power law models become linear when we apply the logarithm transformation to both variables

A power law model is in the form:

Take the logarithm of both sides of the equation and you see that you get:

Look carefully and you see that the power p in the power law becomes the slope of the straight line that links to

EXAMPLE 2 – PREDICTING BRAIN WEIGHT

By taking another look at the brain weight vs. body weight scatterplot from last class (without the 4 outliers) and comparing it to the power models, it appears to take on the shape of a power model more so than the logarithmic model we had initially said.

This is why the book had transformed by taking the logarithm of both variables as opposed to just the response variable.

EXAMPLE 2 – PREDICTING BRAIN WEIGHT

Using the transformed power model, it is found that the LSRL for the transformed regression is:

This means that the logarithm of brain weight can be predicted from the logarithm of body weight using that least squares regression.

What would you predict Bigfoot’s brain weight to be if his body weight is roughly 280 pounds, or 127 kilograms?

In order to be able to predict the brain weight from body weight without logarithms, we need to “undo” the logarithm transformation using base 10:

Since the original weights were in kilograms, we need to substitute the 127 in for x:

Based on the model, Bigfoot is expected to have a brain weight of 333.7 grams.

EXAMPLE 2 – PREDICTING BRAIN WEIGHT

�̂�=10.2×𝑥 .72

�̂�=10.2×127 .72

grams

EXAMPLE 3 – FISHING TOURNAMENT

Imagine that you have been put in charge of organizing a large fishing tournament in which prizes will be given for the heaviest fish caught. The following table represents the ages, lengths, and weights of 20 fish caught.

Type the values in your calculator and look at its scatterplot What type of graph does it appear to be?

Exponential

EXAMPLE 3 – FISHING TOURNAMENT

Refer to Example 4.9 on p.216 for the rest of the scenario. Although the graph appears exponential, it has already been decided

that the weight can be estimated using the function:

This function is in the form which would be representative of a power function

In order to straighten out the graph you will need to take the logarithm of both variables

Since the transformed scatterplot appears to be linear, you can run a regression

EXAMPLE 3 – FISHING TOURNAMENT Recall the power model was

Thus, the equation for the LSRL for the transformed power model is:

or

(which looks like allowing us to run a linear regression)

EXAMPLE 3 – FISHING TOURNAMENT

Run the regression of logy on logx and you see that the least squares regression is:

and

Remember to check the residual plot even though the correlation is virtually 1

EXAMPLE 3 – FISHING TOURNAMENT

The last step is to perform an inverse transformation on the linear regression equation:

This represents the final power equation for

the original data.

EXAMPLE 3 – FISHING TOURNAMENT

Suppose your catch measured 36 centimeters. Predict the weight.

702.08