1. Copyright © 2014, 2011 Pearson Education, Inc. 1 Chapter 6 Association between Quantitative Variables

2. Copyright © 2014, 2011 Pearson Education, Inc. 2 6.1 Scatterplots Is household natural gas consumption associated with climate?  Annual household natural gas consumption measured in thousands of cubic feet (MCF)  Climate as measured by the National Weather Service using heating degree days (HDD)

3. Copyright © 2014, 2011 Pearson Education, Inc. 3 6.1 Scatterplots Association between Numerical Variables  A graph displaying pairs of values as points on a two-dimensional grid  The explanatory variable is placed on the x-axis  The response variable is placed on the y-axis

4. Copyright © 2014, 2011 Pearson Education, Inc. 4 6.1 Scatterplots Scatterplot of Natural Gas Consumption (y) versus Heating Degree-Days (x)

5. Copyright © 2014, 2011 Pearson Education, Inc. 5 6.2 Association in Scatterplots Visual Test for Association  Compare the original scatterplot to others that randomly match the coordinates  If you can pick the original out as having a pattern, then there is an association

6. Copyright © 2014, 2011 Pearson Education, Inc. 6 6.2 Association in Scatterplots Describing Association 1. Direction. Does it trend up or down? 2. Curvature. Is the pattern linear or curved? 3. Variation. Are the points tightly clustered around the trend? 4. Outliers. Is there something unexpected?

7. Copyright © 2014, 2011 Pearson Education, Inc. 7 6.2 Association in Scatterplots Gas Consumption vs. Heating Degree Days 1. Direction: Positive. 2. Curvature: Linear. 3. Variation: Considerable scatter. 4. Outliers: None apparent.

8. Copyright © 2014, 2011 Pearson Education, Inc. 8 6.3 Measuring Association Covariance  A measure that quantifies the linear association  Depends on units of measurement and is therefore difficult to interpret

9. Copyright © 2014, 2011 Pearson Education, Inc. 9 6.3 Measuring Association Calculating the Covariance (for n = 6 homes)

10. Copyright © 2014, 2011 Pearson Education, Inc. 10 6.3 Measuring Association Correlation (r)  Standardized measure of the strength of the linear association (has no units)  Always between -1 and +1  Easy to interpret

11. Copyright © 2014, 2011 Pearson Education, Inc. 11 6.3 Measuring Association Gas Consumption and Heating Degree Days Cov (HDD, Gas) = 56,308.9 HDD X MCF Corr (HDD, Gas) = 0.58 The association is positive and moderate.

12. Copyright © 2014, 2011 Pearson Education, Inc. 12 6.3 Measuring Association Scatterplot for r = 1

13. Copyright © 2014, 2011 Pearson Education, Inc. 13 6.3 Measuring Association Scatterplot for r = -0.95

14. Copyright © 2014, 2011 Pearson Education, Inc. 14 6.3 Measuring Association Scatterplot for r = 0.75

15. Copyright © 2014, 2011 Pearson Education, Inc. 15 6.3 Measuring Association Scatterplot for r = -0.50

16. Copyright © 2014, 2011 Pearson Education, Inc. 16 6.3 Measuring Association Scatterplot for r = 0

17. Copyright © 2014, 2011 Pearson Education, Inc. 17 6.3 Measuring Association Correlation Size  Depends on context  Correlations between macroeconomic variables often approach 1  Smaller correlations are typical for behavioral data

18. Copyright © 2014, 2011 Pearson Education, Inc. 18 6.3 Measuring Association Macroeconomic Variables

19. Copyright © 2014, 2011 Pearson Education, Inc. 19 6.3 Measuring Association Consumer Behavior Variables

20. Copyright © 2014, 2011 Pearson Education, Inc. 20 6.4 Summarizing Association with a Line Expressed using z-scores Slope-Intercept Form with and

21. Copyright © 2014, 2011 Pearson Education, Inc. 21 6.4 Summarizing Association with a Line Line Relating Gas Consumption (y) to Heating Degree Days (x)

22. Copyright © 2014, 2011 Pearson Education, Inc. 22 6.4 Summarizing Association with a Line Lines and Prediction  Use the correlation line to customize an ad for estimated savings from insulation based on climate.  For a home in a cold climate (HDD = 8,800), the predicted gas consumption is 141 MCF.  At $10 / MCF, the predicted cost is $1,410.  Assuming that insulation saves 30% on gas bill, estimated savings is $423.

23. Copyright © 2014, 2011 Pearson Education, Inc. 23 6.4 Summarizing Association with a Line Nonlinear Patterns  If the association is not linear, a line may be a poor summary of the pattern  Covariance and correlation measure only linear association  Inspect the scatterplot before relying on these statistics to measure association

24. Copyright © 2014, 2011 Pearson Education, Inc. 24 6.5 Spurious Correlation Lurking Variables  Scatterplots and correlation reveal association, not causation  Spurious correlations result from underlying lurking variables

25. Copyright © 2014, 2011 Pearson Education, Inc. 25 6.5 Spurious Correlation Checklist: Covariance and Correlation  Numerical variables  No obvious lurking variables  Linear  Outliers

26. Copyright © 2014, 2011 Pearson Education, Inc. 26 4M Example 6.1: LOCATING A NEW STORE Motivation Is it better to locate a new retail outlet far from competing stores?

27. Copyright © 2014, 2011 Pearson Education, Inc. 27 4M Example 6.1: LOCATING A NEW STORE Method Is there an association between sales at the retail outlets and distance to nearest competitor? For 55 stores in the chain, data are gathered for total sales in the prior year and distance in miles from the nearest competitor.

28. Copyright © 2014, 2011 Pearson Education, Inc. 28 4M Example 6.1: LOCATING A NEW STORE Mechanics

29. Copyright © 2014, 2011 Pearson Education, Inc. 29 4M Example 6.1: LOCATING A NEW STORE Mechanics Compute the correlation between sales and distance to be r = 0.741

30. Copyright © 2014, 2011 Pearson Education, Inc. 30 4M Example 6.1: LOCATING A NEW STORE Message The data show a strong, positive linear association between distance to the nearest competitor and sales. It is better to locate a new store far from its competitors.

31. Copyright © 2014, 2011 Pearson Education, Inc. 31 Best Practices  To understand the relationship between two numerical variables, start with a scatterplot.  Look at the plot, look at the plot, look at the plot.  Use clear labels for the scatterplot.

32. Copyright © 2014, 2011 Pearson Education, Inc. 32 Best Practices (Continued)  Describe a relationship completely.  Consider the possibility of lurking variables.  Use a correlation to quantify the association between two numerical variables that are linearly related.

33. Copyright © 2014, 2011 Pearson Education, Inc. 33 Pitfalls  Don’t use the correlation if data are categorical.  Don’t treat association and correlation as causation.  Don’t assume that a correlation of zero means that the variables are not associated.  Don’t assume that a correlation near -1 or +1 means near perfect association.