1. Chapter 141 Describing Relationships: Scatterplots and Correlation

2. Chapter 142 Thought Question 1 For all cars manufactured in the U.S., there is a positive correlation between the size of the engine and horsepower. There is a negative correlation between the size of the engine and gas mileage. What does it mean for two variables to have a positive correlation or a negative correlation?

3. Chapter 143 Thought Question 2 What type of correlation would the following pairs of variables have – positive, negative, or none? 1. Temperature during the summer and electricity bills 2. Temperature during the winter and heating costs 3. Number of years of education and height 4. Frequency of brushing and number of cavities 5. Number of churches and number of bars in cities in your state 6. Height of husband and height of wife

4. Chapter 144 Thought Question 3 Consider the two scatterplots below. How does the outlier impact the correlation for each plot? –does the outlier increase the correlation, decrease the correlation, or have no impact?

5. Chapter 145 Statistical versus Deterministic Relationships u Distance versus Speed (when travel time is constant). u Income (in millions of dollars) versus total assets of banks (in billions of dollars).

6. Chapter 146 Distance versus Speed u Distance = Speed  Time u Suppose time = 1.5 hours u Each subject drives a fixed speed for the 1.5 hrs –speed chosen for each subject varies from 10 mph to 50 mph u Distance does not vary for those who drive the same fixed speed u Deterministic relationship

7. Chapter 147 Income versus Assets u Income = a + b  Assets u Assets vary from 3.4 billion to 49 billion u Income varies from bank to bank, even among those with similar assets u Statistical relationship

8. Chapter 148 Strength and Statistical Significance u A strong relationship seen in the sample may indicate a strong relationship in the population. u The sample may exhibit a strong relationship simply by chance and the relationship in the population is not strong or is zero. u The observed relationship is considered to be statistically significant if it is stronger than a large proportion of the relationships we could expect to see just by chance.

9. Chapter 149 Warnings about Statistical Significance u “Statistical significance” does not imply the relationship is strong enough to be considered “practically important”. u Even weak relationships may be labeled statistically significant if the sample size is very large. u Even very strong relationships may not be labeled statistically significant if the sample size is very small.

10. Chapter 1410 Linear Relationship Some relationships are such that the points of a scatterplot tend to fall along a straight line — linear relationship

11. Chapter 1411 Examples of Relationships

12. Chapter 1412 Measuring Strength & Direction of a Linear Relationship u How closely does a non-horizontal straight line fit the points of a scatterplot? u The correlation coefficient (often referred to as just correlation): r –measure of the strength of the relationship: the stronger the relationship, the larger the magnitude of r. –measure of the direction of the relationship: positive r indicates a positive relationship, negative r indicates a negative relationship. Click for Computation

13. Chapter 1413 Correlation Coefficient u special values for r:  a perfect positive linear relationship would have r = +1  a perfect negative linear relationship would have r = -1  if there is no linear relationship, or if the scatterplot points are best fit by a horizontal line, then r = 0  Note: r must be between -1 and +1, inclusive u r > 0: as one variable changes, the other variable tends to change in the same direction u r < 0: as one variable changes, the other variable tends to change in the opposite direction Plot

14. Chapter 1414 Examples of Correlations u Husband’s versus Wife’s ages v r =.94 u Husband’s versus Wife’s heights v r =.36 u Professional Golfer’s Putting Success: Distance of putt in feet versus percent success v r = -.94 Plot Click for Graphical Examples

15. Chapter 1415 Not all Relationships are Linear Miles per Gallon versus Speed u Linear relationship? MPG = a + b  Speed u Speed chosen for each subject varies from 20 mph to 60 mph u MPG varies from trial to trial, even at the same speed u Statistical relationship

16. Chapter 1416 Not all Relationships are Linear Miles per Gallon versus Speed u Curved relationship (r is misleading) u Speed chosen for each subject varies from 20 mph to 60 mph u MPG varies from trial to trial, even at the same speed u Statistical relationship

17. Chapter 1417 Problems with Correlations u Outliers can inflate or deflate correlations u Groups combined inappropriately may mask relationships (a third variable) –groups may have different relationships when separated Plot

18. Chapter 1418 Outliers and Correlation For each scatterplot above, how does the outlier affect the correlation? AB A: outlier decreases the correlation B: outlier increases the correlation

19. Chapter 1419 Price of Books versus Size u Relationship between price of books and the number of pages? u Positive? u Look at paperbacks: u Look at hard covers: u All books together: u Overall correlation is Negative!

20. Chapter 1420 Key Concepts u Statistical vs. Deterministic Relationships u Statistically Significant Relationship u Strength of Linear Relationship u Direction of Linear Relationship u Correlation Coefficient u Problems with Correlations

21. Chapter 1421 Correlation Calculation u Suppose we have data on variables X and Y for n individuals: x 1, x 2, …, x n and y 1, y 2, …, y n u Each variable has a mean and std dev:

22. Chapter 1422 Case Study Per Capita Gross Domestic Product and Average Life Expectancy for Countries in Western Europe

23. Chapter 1423 Case Study CountryPer Capita GDP (x)Life Expectancy (y) Austria21.477.48 Belgium23.277.53 Finland20.077.32 France22.778.63 Germany20.877.17 Ireland18.676.39 Italy21.578.51 Netherlands22.078.15 Switzerland23.878.99 United Kingdom21.277.37

24. Chapter 1424 Case Study xy 21.477.48-0.078-0.3450.027 23.277.531.097-0.282-0.309 20.077.32-0.992-0.5460.542 22.778.630.7701.1020.849 20.877.17-0.470-0.7350.345 18.676.39-1.906-1.7163.271 21.578.51-0.0130.951-0.012 22.078.150.3130.4980.156 23.878.991.4891.5552.315 21.277.37-0.209-0.4830.101 = 21.52 = 77.754 sum = 7.285 s x =1.532s y =0.795

25. Chapter 1425 Case Study There is a strong, positive linear relationship between Per Capita GDP (x) and Life Expectancy (y). Return to Slide 12

26. Chapter 1426 Examples of Correlations Return to Slide 14