Power Models

1. Power Models Presentation 2-10

2. Power Models • Consider the following dataset about the cost of pizzas from Joanies Pizzaria. • Of course, we start with a scatterplot.

3. Power Models • From the scatterplot, we probably suspect it is not linear. • It may be exponential or something else.

4. Power Scatterplots • Power models appear when each x is raised to a constant power. • For powers between 0 and 1, the model is logarithmic.

5. Power Situations • Area and Volume • Comparing length to area (2nd power) or volume (3rd power) • Such as the pizza problem or relating foot length to weight (as in most real regressions, there are outliers). • Predicting the height of a tree from its cross sectional area of its trunk.

6. Power Transformation • The trick is to change the data such that it is linear. • This can be done by taking the logarithm of both the explanatory and response variables. • Remember properties of logarithms (when you multiply powers, you add (linear) exponents • So, essentially, you change the x values and y values to exponents • Now look at the new scatterplot of (log x, log y)

7. Power Transformation • Now look at the new scatterplot of (log x, log y) • This scatterplot looks more centered about a line as opposed to a curve. • Now that it is linear, we do a least-squares regression line on the transformed (log x, log y)

8. Power Transformation • The r-squared value looks good (0.9797) • The regression line (LSRL) is given as: y = 1.3389x – 0.1148 • This is not quite right as it should be: log y = 1.3389*log x – 0.1148

9. Power Transformation • Now to solve the equation for y, so we don’t have to deal with the logarithm. • Start by exponentiating, then using properties of logarithms to solve for y-hat.

10. Power Transformation • This could also be done using ln or natural logarithms. • The only difference would be base e instead of base 10.

11. Power Transformation • This concludes the presentation.