Chapter 4: More on Two-Variable Data

1. Chapter 4: More on Two-Variable Data Part 2

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

3. 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.

4. 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:

5. Example 2 – Predicting Brain Weight • 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. grams

6. 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

7. 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

8. 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)

9. 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

10. 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.

11. Example 3 – Fishing Tournament • Suppose your catch measured 36 centimeters. Predict the weight. 702.08