1. Correlation and Regression
Chapter 9
Correlation and Regression
Larson/Farber 4th ed.

2. Correlation Correlation A relationship between two variables.
The data can be represented by ordered pairs (x, y)
x is the independent (or explanatory) variable
y is the dependent (or response) variable
Larson/Farber 4th ed.

3. 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
Larson/Farber 4th ed.

4. 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
Larson/Farber 4th ed.

5. Example: Constructing a Scatter Plot
A marketing manager conducted a study to determine whether there is a linear relationship between money spent on advertising and company sales. The data are shown in the table. Display the data in a scatter plot and determine whether there appears to be a positive or negative linear correlation or no linear correlation.
Advertising expenses, ($1000), x
Company sales ($1000), y
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
Larson/Farber 4th ed.

6. Solution: Constructing a Scatter Plotx y
Advertising expenses
(in thousands of dollars)
Company sales
Appears to be a positive linear correlation. As the advertising expenses increase, the sales tend to increase.
Larson/Farber 4th ed.

7. Example: Constructing a Scatter Plot Using Technology
Old Faithful, located in Yellowstone National Park, is the world’s most famous geyser. The duration (in minutes) of several of Old Faithful’s eruptions and the times (in minutes) until the next eruption are shown in the table. Using a TI-83/84, display the data in a scatter plot. Determine the type of correlation.
Duration x
Time, y
1.8 56
3.78 79
1.82 58
3.83 85
1.9 62
3.88 80
1.93
4.1 89
1.98 57
4.27 90
2.05
4.3
2.13 60
4.43
2.3
4.47 86
2.37 61
4.53
2.82 73
4.55
3.13 76
4.6 92
3.27 77
4.63 91
3.65
Larson/Farber 4th ed.

8. Solution: Constructing a Scatter Plot Using Technology
Enter the x-values into list L1 and the y-values into list L2.
Use Stat Plot to construct the scatter plot.
STAT > Edit…
STATPLOT 1 5 50
100
From the scatter plot, it appears that the variables have a positive linear correlation.
Larson/Farber 4th ed.

9. 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 formula for r is
The population correlation coefficient is represented by ρ (rho).
n is the number of data pairs
Larson/Farber 4th ed.

10. 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
Larson/Farber 4th ed.

11. 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
Larson/Farber 4th ed.

12. Calculating a Correlation Coefficient
In Words In Symbols
Find the sum of the x-values.
Find the sum of the y-values.
Multiply each x-value by its corresponding y-value and find the sum.
Larson/Farber 4th ed.

13. Calculating a Correlation Coefficient
In Words In Symbols
Square each x-value and find the sum.
Square each y-value and find the sum.
Use these five sums to calculate the correlation coefficient.
Larson/Farber 4th ed.

14. Example: Finding the Correlation Coefficient
Calculate the correlation coefficient for the advertising expenditures and company sales data. What can you conclude?
Advertising expenses, ($1000), x
Company sales ($1000), y
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
Larson/Farber 4th ed.

15. Solution: Finding the Correlation Coefficientx y xy x2 y2
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
540
5.76
50,625
294.4
2.56
33,856
440 4
48,400
624
6.76
57,600
252
1.96
32,400
294.4
2.56
33,856
372 4
34,596
473
4.84
46,225
Σx = 15.8
Σy = 1634
Σxy =
Σx2 = 32.44
Σy2 = 337,558
Larson/Farber 4th ed.

16. Solution: Finding the Correlation Coefficient
Σx = 15.8
Σy = 1634
Σxy =
Σx2 = 32.44
Σy2 = 337,558
r ≈ suggests a strong positive linear correlation. As the amount spent on advertising increases, the company sales also increase.
Larson/Farber 4th ed.

17. Example: Using Technology to Find a Correlation Coefficient
Use a technology tool to calculate the correlation coefficient for the Old Faithful data. What can you conclude?
Duration x
Time, y
1.8 56
3.78 79
1.82 58
3.83 85
1.9 62
3.88 80
1.93
4.1 89
1.98 57
4.27 90
2.05
4.3
2.13 60
4.43
2.3
4.47 86
2.37 61
4.53
2.82 73
4.55
3.13 76
4.6 92
3.27 77
4.63 91
3.65
Larson/Farber 4th ed.

18. 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.
Larson/Farber 4th ed.

19. Using a Table to Test a Population Correlation Coefficient ρ
Once the sample correlation coefficient r has been calculated, we need to determine whether there is enough evidence to decide that the population correlation coefficient ρ is significant at a specified level of significance.
Use Table 11 in Appendix B.
If |r| is greater than the critical value, there is enough evidence to decide that the correlation coefficient ρ is significant.
Larson/Farber 4th ed.

20. Using a Table to Test a Population Correlation Coefficient ρ
Determine whether ρ is significant for five pairs of data (n = 5) at a level of significance of α = 0.01.
If |r| > 0.959, the correlation is significant. Otherwise, there is not enough evidence to conclude that the correlation is significant.
level of significance
Number of pairs of data in sample
Larson/Farber 4th ed.

21. Using a Table to Test a Population Correlation Coefficient ρ
In Words In Symbols
Determine the number of pairs of data in the sample.
Specify the level of significance.
Find the critical value.
Determine n.
Identify .
Use Table 11 in Appendix B.
Larson/Farber 4th ed.

22. Using a Table to Test a Population Correlation Coefficient ρ
In Words In Symbols
Decide if the correlation is significant.
Interpret the decision in the context of the original claim.
If |r| > critical value, the correlation is significant. Otherwise, there is not enough evidence to support that the correlation is significant.
Larson/Farber 4th ed.

23. Example: Using a Table to Test a Population Correlation Coefficient ρ
Using the Old Faithful data, you used 25 pairs of data to find r ≈ Is the correlation coefficient significant? Use α = 0.05.
Duration x
Time, y
1.8 56
3.78 79
1.82 58
3.83 85
1.9 62
3.88 80
1.93
4.1 89
1.98 57
4.27 90
2.05
4.3
2.13 60
4.43
2.3
4.47 86
2.37 61
4.53
2.82 73
4.55
3.13 76
4.6 92
3.27 77
4.63 91
3.65
Larson/Farber 4th ed.

24. Solution: Using a Table to Test a Population Correlation Coefficient ρ
There is enough evidence at the 5% level of significance to conclude that there is a significant linear correlation between the duration of Old Faithful’s eruptions and the time between eruptions.
Larson/Farber 4th ed.

25. Hypothesis Testing for a Population Correlation Coefficient ρ
A hypothesis test can also be used to determine whether the sample correlation coefficient r provides enough evidence to conclude that the population correlation coefficient ρ is significant at a specified level of significance.
A hypothesis test can be one-tailed or two-tailed.
Larson/Farber 4th ed.

26. Hypothesis Testing for a Population Correlation Coefficient ρ
Left-tailed test
Right-tailed test
Two-tailed test
H0: ρ  0 (no significant negative correlation) Ha: ρ < 0 (significant negative correlation)
H0: ρ  0 (no significant positive correlation) Ha: ρ > 0 (significant positive correlation)
H0: ρ = 0 (no significant correlation) Ha: ρ  0 (significant correlation)
Larson/Farber 4th ed.

27. The t-Test for the Correlation Coefficient
Can be used to test whether the correlation between two variables is significant.
The test statistic is r
The standardized test statistic
follows a t-distribution with d.f. = n – 2.
In this text, only two-tailed hypothesis tests for ρ are considered.
Larson/Farber 4th ed.

28. Using the t-Test for ρ In Words In Symbols
State the null and alternative hypothesis.
Specify the level of significance.
Identify the degrees of freedom.
Determine the critical value(s) and rejection region(s).
State H0 and Ha.
Identify .
d.f. = n – 2.
Use Table 5 in Appendix B.
Larson/Farber 4th ed.

29. Using the t-Test for ρ In Words In Symbols
Find the standardized test statistic.
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If t is in the rejection region, reject H0. Otherwise fail to reject H0.
Larson/Farber 4th ed.

30. Example: t-Test for a Correlation Coefficient
Previously you calculated r ≈ Test the significance of this correlation coefficient. Use α = 0.05.
Advertising expenses, ($1000), x
Company sales ($1000), y
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
Larson/Farber 4th ed.

31. Solution: t-Test for a Correlation Coefficient
H0:
Ha:
 
d.f. =
Rejection Region:
ρ = 0
ρ ≠ 0
Test Statistic:
0.05
8 – 2 = 6
Decision:
Reject H0
At the 5% level of significance, there is enough evidence to conclude that there is a significant linear correlation between advertising expenses and company sales. t
-2.447
0.025
2.447
-2.447
2.447
5.478
Larson/Farber 4th ed.

32. 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?
Larson/Farber 4th ed.

33. 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?
Larson/Farber 4th ed.

34. Section 9.2
Linear Regression
Larson/Farber 4th ed.

35. Regression lines
After verifying that the linear correlation between two variables is significant, next we determine the equation of the line that best models the data (regression line).
Can be used to predict the value of y for a given value of x. x y
Larson/Farber 4th ed.

36. Residuals
Residual
The difference between the observed y-value and the predicted y-value for a given x-value on the line.
For a given x-value,
di = (observed y-value) – (predicted y-value) x y
}d1
}d2
d3{
d4{
}d5
d6{
Predicted y-value
Observed y-value
Larson/Farber 4th ed.

37. Regression Line Regression line (line of best fit)
The line for which the sum of the squares of the residuals is a minimum.
The equation of a regression line for an independent variable x and a dependent variable y is
ŷ = mx + b
y-intercept
Predicted y-value for a given x-value
Slope
Larson/Farber 4th ed.

38. The Equation of a Regression Line
ŷ = mx + b where
is the mean of the y-values in the data
is the mean of the x-values in the data
The regression line always passes through the point
Larson/Farber 4th ed.

39. Example: Finding the Equation of a Regression Line
Find the equation of the regression line for the advertising expenditures and company sales data.
Advertising expenses, ($1000), x
Company sales ($1000), y
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
Larson/Farber 4th ed.

40. Solution: Finding the Equation of a Regression Line
Recall from section 9.1: x y xy x2 y2
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
540
5.76
50,625
294.4
2.56
33,856
440 4
48,400
624
6.76
57,600
252
1.96
32,400
294.4
2.56
33,856
372 4
34,596
473
4.84
46,225
Σx = 15.8
Σy = 1634
Σxy =
Σx2 = 32.44
Σy2 = 337,558
Larson/Farber 4th ed.

41. Solution: Finding the Equation of a Regression Line
Σx = 15.8
Σy = 1634
Σxy =
Σx2 = 32.44
Σy2 = 337,558
Equation of the regression line
Larson/Farber 4th ed.

42. Solution: Finding the Equation of a Regression Line
To sketch the regression line, use any two x-values within the range of the data and calculate the corresponding y-values from the regression line. x
Advertising expenses
(in thousands of dollars)
Company sales y
Larson/Farber 4th ed.

43. Example: Using Technology to Find a Regression Equation
Use a technology tool to find the equation of the regression line for the Old Faithful data.
Duration x
Time, y
1.8 56
3.78 79
1.82 58
3.83 85
1.9 62
3.88 80
1.93
4.1 89
1.98 57
4.27 90
2.05
4.3
2.13 60
4.43
2.3
4.47 86
2.37 61
4.53
2.82 73
4.55
3.13 76
4.6 92
3.27 77
4.63 91
3.65
Larson/Farber 4th ed.

44. Solution: Using Technology to Find a Regression Equation
100 50 1 5
Larson/Farber 4th ed.

45. Example: Predicting y-Values Using Regression Equations
The regression equation for the advertising expenses (in thousands of dollars) and company sales (in thousands of dollars) data is ŷ = x Use this equation to predict the expected company sales for the following advertising expenses. (Recall from section 9.1 that x and y have a significant linear correlation.)
1.5 thousand dollars
1.8 thousand dollars
2.5 thousand dollars
Larson/Farber 4th ed.

46. Solution: Predicting y-Values Using Regression Equations
ŷ = x
1.5 thousand dollars
ŷ =50.729(1.5) ≈
When the advertising expenses are $1500, the company sales are about $180,155.
1.8 thousand dollars
ŷ =50.729(1.8) ≈
When the advertising expenses are $1800, the company sales are about $195,373.
Larson/Farber 4th ed.

47. Solution: Predicting y-Values Using Regression Equations
2.5 thousand dollars
ŷ =50.729(2.5) ≈
When the advertising expenses are $2500, the company sales are about $230,884.
Prediction values are meaningful only for x-values in (or close to) the range of the data. The x-values in the original data set range from 1.4 to 2.6. So, it would
not be appropriate to use the regression line to predict
company sales for advertising expenditures such as 0.5 ($500) or 5.0 ($5000).
Larson/Farber 4th ed.