More about Relationships between Two Variables

1. More about Relationships between Two Variables

2. Case Study (p. 257)It’s a matter of life and death • How do insurance companies decide how much to charge for life insurance? • Actuary • How much would a 58-year-old expect to pay for such a policy? • A 68-year-old?

3. Chapter 3 Learned to analyze relationships between two quantitative variables that showed a linear pattern Chapter 4 When two-variable data shows a nonlinear relationship…

4. 4.1 Transforming to Achieve Linearity

5. Example 4.1(p. 259): Modeling mammal brain weight versus body weight Figure 4.1

7. Transforming or re-expressing the data.

8. Regression • Regression techniques in this course are those of LINEAR regression • If data set does not appear to be linear, we look for a transformation that will linearize the data • There are techniques to find a best-fit curve for nonlinear data – we just won’t be using them in this course

9. Example 4.2: Fishing tournamentTransforming data with powers Original data Transformed data

10. Original data points, with transformed model equation Many different transformations of data are possible, we will concentrate on power and logarithmic transformations.

11. Intro to Nonlinear Relationships • Data that displays a curved pattern can be modeled by a number of different functions. • Two of the most common nonlinear models are exponential (y=abx) and power (y=axb). • Our goal is to fit a model to curved data so that we can make predictions like we did in Chapter 3 • The only statistical tool we have to fit a model is the least-squares regression model • So, in order to find a model for curved data, we must first “straighten it out”

12. Example 4.3: A country’s gross domestic product and life expectancy

13. Ladder of Powers Long and tedious process.

14. Linear Growth: increases by a fixed amount in each equal time period successive terms are related by addition Exponential Growth: Increases by a fixed percent of the previous total in each equal time period occurs when a variable is multiplied by a fixed number in each equal time period successive terms are related by multiplication Linear vs. Exponential Growth

15. Properties of Logarithms

16. Example 4.5: Moore’s law and computer chipsExponential growth

24. Homework • P. 276 – 278 • #4.6 and 4.8

25. p. 276 4.6 Gypsy moths (b) (a) (d) (c)

26. (e) (f) (g) The residual plot shows no clear pattern, so the linear regression model on the transformed data is appropriate.

27. (h)

28. (h) The exponential model provides and excellent fit. (i) Predicted number of acres defoliated is 10,722,597.42.

29. p. 277 4.8 Gun violence (a) (b)

30. (b) (b) (c) (b) 35,184,372,088,832 About 35 trillion deaths from gun violence. This is clearly a mistake.