Analyzing Two-Variable Data

1. Analyzing Two-Variable Data Lesson 2.8 Fitting Models to Curved Relationships 2

2. Fitting Models to Curved Relationships • Use technology to calculate quadratic models for curved relationships, then calculate and interpret residuals using the model. • Use technology to calculate exponential models for curved relationships, then calculate and interpret residuals using the model. • Use residual plots to determine the most appropriate model.

3. Fitting Models to Curved Relationships When the association between two quantitative variables is linear, we use a least-squares regression line to model the relationship. When an association is nonlinear, we can use technology to calculate other types of models. In some cases, we can use a quadratic model to fit a nonlinear association between two quantitative variables. Quadratic Model A quadratic model is a model in the form yˆ = ax2 +bx +c . The graph of a quadratic model is a parabola.

4. Fitting Models to Curved Relationships The scatterplot below shows the relationship between passing yards and age for 32 quarterbacks in the 2010 NFL season.

5. Fitting Models to Curved Relationships The association between passing yards and age is clearly nonlinear. Passing yards tend to be lower for younger and older quarterbacks and higher for “middle-aged” quarterbacks.

6. Pitching for WHIP?Calculating and using a quadratic model Baseball pitchers try to prevent hitters from reaching base. One way to measure the effectiveness of a pitcher is WHIP, which is calculated by the formula: . The table shows the age and WHIP for all 17 years of Bob Gibson’s MLB hall-of-fame career. PROBLEM: • Calculate a quadratic model for these data using age as the explanatory variable. • Sketch the scatterplot, along with the quadratic model. • Calculate and interpret the residual for observation when Bob was 24 years old.

7. Pitching for WHIP?Calculating and using a quadratic model (a) Calculate a quadratic model for these data using age as the explanatory variable. (b) Sketch the scatterplot, along with the quadratic model. (c) Calculate and interpret the residual for observation when Bob was 24 years old. . Bob’s WHIP when he was 24 years old was 0.070 greater than predicted by the quadratic model with x = age.

8. Fitting Models to Curved Relationships We can also use an exponential model to fit a nonlinear association between two quantitative variables. Exponential Model • An exponential model is a model in the form y^ = a(b)x where b > 0. • If b > 1, the graph will show exponential growth. • If 0 < b < 1, the graph will show exponential decay.

9. Fitting Models to Curved Relationships Gordon Moore, one of the founders of Intel Corporation, predicted in 1965 that the number of transistors on an integrated circuit chip would double every 18 months. This is Moore’s law, one way to measure the revolution in computing. Here is a scatterplot showing the number of transistors for Intel microprocessors and the number of years since 1970.

10. Does health predict wealth?Calculating and using an exponential model Do healthier countries tend to be wealthier countries? A random sample of 20 countries was obtained from gapminder.org. The income per person (US dollars) and life expectancy at birth (years) were recorded. PROBLEM: (a) Calculate an exponential model for these data using life expectancy as the explanatory variable. (b) Sketch the scatterplot, along with the exponential model. (c) Calculate and interpret the residual for Mexico.

11. Does health predict wealth?Calculating and using an exponential model (a) Calculate an exponential model for these data using life expectancy as the explanatory variable. (b) Sketch the scatterplot, along with the exponential model. (c) Calculate and interpret the residual for Mexico. The annual income per person for Mexico was $2480 less than predicted by the exponential model with x = life expectancy.

12. Fitting Models to Curved Relationships In some cases, it is hard to tell if a quadratic model or an exponential model would be better to use. To decide, look at the residual plots for both models and choose the model with the residual plot that has the most random scatter.

13. Black gold in North Dakota?Choosing a model PROBLEM: Texas is the state that produces the most oil in the United States, but which state is the second-largest producer? In 2014, North Dakota produced 397 million barrels of oil, making it the second-largest producer of oil in the United States. A scatterplot of the annual oil production (million barrels) versus year (years since 2000) for the state of North Dakota shows a distinct nonlinear relationship.

14. Black gold in North Dakota?Choosing a model An exponential model and a quadratic model were calculated for the relationship between years since 2000 and annual oil production. The residual plots for these models are shown below. Based on the residual plots, which model is more appropriate? Explain. Because the residual plot for the quadratic model seems more randomly scattered, the quadratic model is more appropriate for these data than the exponential model.

15. LESSON APP 2.8 How does life insurance work? Many adults try to protect their families by buying life insurance. The policyholder makes regular payments (premiums) to the insurance company in return for the coverage. When the insured person dies, a payment is made to designated family members or other beneficiaries.

16. LESSON APP 2.8 How does life insurance work? How do insurance companies decide how much to charge for life insurance? They rely on a staff of highly trained actuaries—people with expertise in probability, statistics, and advanced mathematics—to determine premiums. If someone wants to buy life insurance, the premium will depend on the type and amount of the policy as well as on personal characteristics like age, gender, and health status. The table shows monthly premiums (in dollars) for a 10-year term-life insurance policy worth $1,000,000 for people of various ages (in years).

17. LESSON APP 2.8 How does life insurance work? • Calculate a quadratic model for these data. • Calculate an exponential model for these data.

18. LESSON APP 2.8 How does life insurance work? • Which model is more appropriate? Justify your answer. • Using your chosen model, calculate and interpret the residual for the 65-year-old.

19. Fitting Models to Curved Relationships • Use technology to calculate quadratic models for curved relationships, then calculate and interpret residuals using the model. • Use technology to calculate exponential models for curved relationships, then calculate and interpret residuals using the model. • Use residual plots to determine the most appropriate model.