+ Chapter 12 Section 2 Transforming to Achieve Linearity

+ Students will be able to: Use transformations involving powers and roots to achieve linearity for a relationship between two variables. Use transformations involving logarithms to achieve linearity for a relationship between two variables. Make predictions from a least-squares regression line involving transformed data. Determine which of several transformations does a better job of producing a linear relationship.

+ Transforming to Achieve Linearity When two-variable data show a curved relationship, we must develop new techniques for finding an appropriate model. Once the data have been transformed to achieve linearity, we can use least-squares regression to generate a useful model for making predictions. If the conditions for regression inference are met, we can estimate or test a claim about the slope of the population regression line using the transformed data.

+ Transforming to Achieve Linearity Two types of transforming we will use: logarithm and square root. When we apply a function such as one of these, this is called transforming the data. Transforming data amounts to changing the scale of measurement that was used when the data were collected. Linear transformations cannot straighten out curved relationship between two variables. This is why we resort to functions that are not linear.

+ Transforming with Powers and Roots Power model – dealing with circles, fish, people. We expect area to go up with the square of a dimension such as diameter or height. Volume should go up with the cube of a linear dimension. Although a power model of the form y = ax b describes the relationship between x and y in each of these settings, there is a linear relationship between x p and y. Take a look at the example on pp and 770

+ Transforming to Achieve Linearity When experience or theory suggests that the relationship between two variables is described by a power model of the form y = ax b, you now have two strategies for transforming the data to achieve linearity Raise the values of the explanatory variable x to the p power and plot the points (x p,y) Take the pth root for the values of the response variable y and plot the points (x, square root p of y). If we do not know what power to choose then guess and test until you find a transformation that works.

+ Transforming with Logarithms Sometimes not all curved relationships are described by power models but by a logarithm (base 10) – read paragraph on pp. 771 under heading of logarithms for clarification. Exponential model: read paragraph on pp. 771 Take a look at example on pp

+ Transforming with Logarithms Take a look at the rest of the examples for clarification I would work the odd problems in the practice section for further understanding.