12.2 TRANSFORMING TO ACHIEVE LINEARITY To use transformations involving powers, roots, and logarithms to find a power or exponential model that describes the relationship between two variables and determine which is best to use given a situation. Why extrapolation cannot be trusted.

HOW TO MAKE A RESIDUAL PLOT IN YOUR CALCULATOR

REMEMBER Certain methods that we use to discuss and describe data can only be used with linear data. Non-Linear Data: data that forms a curve in the scatterplot How do we know if data is non-linear?  We might be able to use the scatterplot to look at the curve. However, we better check! The residual plot shows us a pattern.

EXAMPLE Several students noticed that some cars have tinted windows and that some cars appear to have darker tints than others. In an attempt to determine just how much light the tinting material lets into the car, the students decided do a study. They obtained several sheets of tinting material, a flashlight and a light-intensity meter. The scatterplot, residual plot and LINEAR regression equation for the number of sheets and the light-intensity factor are shown on the next slide.

Since the original data is non-linear (curve in the residual plot), the linear model is not appropriate (therefore, r 2 should not be used). Solution: TRANSFORM THE DATA TO MAKE IT LINEAR

MOST COMMON FORMS OF TRANSFORMATION

HOW TO TRANSFORM NON-LINEAR DATA

EXAMPLE The tinting sheets and light intensity data is exponential. Transform the data. Sketch the new scatterplot and here (label axes appropriately) and write the correct (transformed) equation for the line of best fit.

WHAT DO I NEED TO KNOW ABOUT NON-LINEAR DATA? Notice when data is non-linear Identify the correct re-expression technique Rename the line of best fit using the new x and/or y Make Predictions using algebra, as necessary Adjust interpretations and conclusions using new x and/or y

EXAMPLE CONTINUED 1. Interpret the slope of the line for the transformed tinting sheets data in context. 2. The r 2 value for this transformed data was 99.3%. Interpret this value in context. 3. Use the equation from the previous page to predict the light intensity factor for 7 sheets.

HOW DO I KNOW WHICH METHOD IS BEST? - Use residual plots!  Remember: you want a relatively even scatter above and below the residual=0 line

PRACTICE

12.2 TRANSFORMING TO ACHIEVE LINEARITY To use transformations involving powers, roots, and logarithms to find a power or exponential model that describes the relationship between two variables and determine which is best to use given a situation.

HW CHECK & WARMUP What is the appropriate transformation method for linearizing data that has each of the following relationships? Exponential? Logarithmic? Power?

AP-STYLE FRQ Data was collected to determine if the length of an alligator can be used to predict the alligator’s weight. In order to linearize the data, a statistician took the logarithm of the weights and achieved the following residual plot. 1. Did the statistician use an appropriate re-expression technique? 2. What was the form of the original data (before transformation)?

VIDEO: TRANSFORMING TO ACHIEVE LINEARITY (TI-84)

TEXTBOOK PRACTICE p. 787 #35, 37, 41

REST OF CLASS: MAKE UP TIME ANY OUTSTANDING HOMEWORK QUIZZES/ TESTS TO RETAKE MISSING PROJECTS STUDY CHAPTERS 3 AND 12.2