1. Copyright © Cengage Learning. All rights reserved.
Correlation
and Regression 8 10
Copyright © Cengage Learning. All rights reserved.

2. Chapter Outline
10.1 Correlation and Regression 10.2 Linear Regression and Coefficient of Determination 10.3 Inferences for Correlation and Regression 10.4 Multiple Regression
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3. Scatter Diagrams and Linear Correlation
Section 10.1
Scatter Diagrams and Linear Correlation
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4. Section 10.1 Objectives
Introduce linear correlation, independent and dependent variables, and the types of correlation
Describe the direction, form, and strength of a relationship displayed in a scatterplot and identify outliers in a scatterplot.
Find and interpret a correlation coefficient
Distinguish between correlation and causation
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5. Correlation Correlation A relationship between two variables
The data can be represented by ordered pairs (x, y)
x is the independent (or explanatory) variable
An explanatory variable may help explain or influence changes in a response variable.
y is the dependent (or response) variable
A response variable measures an outcome of a study.
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6. Correlation A scatter plot
Can be used to determine whether a linear (straight line) correlation exists between two variables
Shows the relationship between two quantitative variables measured on the same individuals
Each individual in the data appears as a point on the graph.
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7. Correlation
A scatter plot can be used to determine whether a linear (straight line) correlation exists between two variables. x 2 4 –2
– 4 y 6
Example: x 1 2 3 4 5 y
– 4
– 2
– 1
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8. Types of Correlation As x increases, y tends to decrease.
As x increases, y tends to increase.
Negative Linear Correlation
Positive Linear Correlation x y x y
No Correlation
Nonlinear Correlation
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9. Sample Correlation Coefficient r

10. Sample Correlation Coefficient r

11. Correlation Coefficient
The range of the correlation coefficient is –1 to 1. -1 1
If r = –1 there is a perfect negative correlation
If r is close to 0 there is no linear correlation
If r = 1 there is a perfect positive correlation
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12. Linear Correlation Strong negative correlationx y x y
r = –0.91
r = 0.88
Strong negative correlation
Strong positive correlation x y x y
r = 0.42
r = 0.07
Weak positive correlation
Nonlinear Correlation
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13. Calculating r
The formula for r is a bit complex. It helps us to see what correlation is, but in practice, you should use your calculator or software to find r.
How to Calculate the Correlation r
Suppose that we have data on variables x and y for n individuals. The values for the first individual are x1 and y1, the values for the second individual are x2 and y2, and so on. The means and standard deviations of the two variables are x-bar and sx for the x-values and y-bar and sy for the y-values.
The correlation r between x and y is:

14. Calculating r: Computational Formula
Correlation coefficient
A measure of the strength and the direction of a linear relationship between two variables.
The symbol r represents the sample correlation coefficient.
A [computational]formula for r is
The population correlation coefficient is represented by ρ (rho).
n is the number of data pairs
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15. Solution: Using Technology to Find a Correlation Coefficient
To calculate r, you must first enter the DiagnosticOn command found in the Catalog menu
STAT > Calc
r ≈ suggests a strong positive correlation.
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16. Guided Practice 1: Scatter Plot via TI-84
Merchandise loss due to shoplifting, damage, and other causes is called shrinkage. Shrinkage is a major concern to retailers. The managers of Terrapin Sportswear believe there is a relationship between shrinkage and number of clerks on duty. To explore this relationship, a random sample of 6 weeks was selected. During each week the staffing level of sales clerks was kept constant and the dollar value of the shrinkage was recorded.
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17. Scatter Plot viaTI-84 Step 1: Input frequency table data
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18. Scatter Plot viaTI-84 Step 2: Stat Plot Menu:
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19. Scatter Plot viaTI-84 Step 3: Zoom Stat
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20. Scatter Plot viaTI-84 Step 4: Trace
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21. Example: Correlation Coefficient/ Hand Calculation
GDP
(trillions of $), x
CO2 emission (millions of metric tons), y
1.6
428.2
3.6
828.8
4.9
1214.2
1.1
444.6
0.9
264.0
2.9
415.3
2.7
571.8
2.3
454.9
358.7
1.5
573.5
Calculate the correlation coefficient for the gross domestic products and carbon dioxide emissions data. What can you conclude?
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22. Example: Hand Calculationy xy x2 y2
1.6
428.2
685.12
2.56
183,355.24
3.6
828.8
12.96
686,909.44
4.9
1214.2
24.01
1,474,281.64
1.1
444.6
489.06
1.21
197,669.16
0.9
264.0
237.6
0.81
69,696
2.9
415.3
8.41
172,474.09
2.7
571.8
7.29
326,955.24
2.3
454.9
5.29
206,934.01
358.7
573.92
128,665.69
1.5
573.5
860.25
2.25
328,902.25
Σx = 23.1
Σy = 5554
Σxy = 15,573.71
Σx2 = 67.35
Σy2 = 3,775,842.76
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23. Example: Hand Calculation
Σy = 5554
Σxy = 15,573.71
Σx2 = 32.44
Σy2 = 3,775,842.76
r ≈ suggests a strong positive linear correlation. As the gross domestic product increases, the carbon dioxide emissions also increase.
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24. Exercise 1: Using Technology to Find a Correlation Coefficient
In one of the Boston city parks, there has been a problem with muggings in the summer months.
A police cadet took a random sample of 10 days (out of the 90-day summer) and compiled the following data. For each day, x represents the number of police officers on duty in the park and y represents the number of reported muggings on that day.
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25. Exercise 1: Using Technology to Find a Correlation Coefficient
Use the Stat Plot function to construct the plot. Enter the x-values into list L1 and the y-values into list L2. What can you conclude?
b) Use the TI-84 to calculate the correlation coefficient for the data. What can you conclude?
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26. Exercise 1: Using Technology to Find a Correlation Coefficient
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27. Exercise 3: Using Technology to Find a Correlation Coefficient
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28. Correlation and Causation
The fact that two variables are strongly correlated does not in itself imply a cause-and-effect relationship between the variables
If there is a significant correlation between two variables, you should consider the following possibilities:
Is there a direct cause-and-effect relationship between the variables?
does x cause y?
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29. Correlation and Causation
Is there a reverse cause-and-effect relationship between the variables?
does y cause x?
Is it possible that the relationship between the variables can be caused by a third variable or by a combination of several other variables?
Is it possible that the relationship between two variables may be a coincidence?
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30. Facts About Correlation
Correlation makes no distinction between explanatory and response variables.
r does not change when we change the units of measurement of x, y, or both.
The correlation r itself has no unit of measurement.
Cautions:
Correlation requires that both variables be quantitative.
Correlation does not describe curved relationships between variables, no matter how strong the relationship is.
Correlation is not resistant. r is strongly affected by a few outlying observations.
Correlation is not a complete summary of two-variable data.

31. Section 10.1 Summary
Introduced linear correlation, independent and dependent variables and the types of correlation
Found a correlation coefficient
Distinguished between correlation and causation
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