1. Section 13.7 Linear Correlation and Regression

2. What You Will Learn Linear Correlation Scatter Diagram
Linear Regression
Least Squares Line

3. Linear Correlation
Linear correlation is used to determine whether there is a linear relationship between two quantities and, if so, how strong the relationship is.

4. Linear Correlation Coefficient
The linear correlation coefficient, r, is a unitless measure that describes the strength of the linear relationship between two variables.
If the value is positive, as one variable increases, the other increases.
If the value is negative, as one variable increases, the other decreases.
The variable, r, will always be a value between –1 and 1 inclusive.

5. Scatter Diagrams
A visual aid used with correlation is the scatter diagram, a plot of points (bivariate data).
The independent variable, x, generally is a quantity that can be controlled.
The dependent variable, y, is the other variable.

6. Scatter Diagrams
The value of r is a measure of how far a set of points varies from a straight line.
The greater the spread, the weaker the correlation and the closer the r value is to 0.
The smaller the spread, the stronger the correlation and the closer the r value is to 1 or –1.

7. Correlation

8. Correlation

9. Linear Correlation Coefficient
The formula to calculate the correlation coefficient (r) is as follows.

10. Example 1: Number of Absences Versus Number of Defective Parts
Egan Electronics provided the following daily records about the number of assembly line workers absent and the number of defective parts produced for 6 days. Determine the correlation coefficient between the number of workers absent and the number of defective parts produced.

11. Example 1: Number of Absences Versus Number of Defective Parts

12. Example 1: Number of Absences Versus Number of Defective Parts
Solution
Here’s the scatter diagram.

14. Example 1: Number of Absences Versus Number of Defective Parts
Solution
Find r.

15. Example 1: Number of Absences Versus Number of Defective Parts
Solution

16. Example 1: Number of Absences Versus Number of Defective Parts
Solution
Since the maximum possible value for r is 1.00, a correlation coefficient of is a strong, positive correlation.
This result implies that, generally, the more assembly line workers absent, the more defective parts produced.

17. Linear Regression
Linear regression is the process of determining the linear relationship between two variables.

18. Linear Regression
The line of best fit (regression line or the least squares line) is the line such that the sum of the squares of the vertical distances from the line to the data points (on a scatter diagram) is a minimum.

19. The Line of Best Fit
The equation of the line of best fit is

20. Example 3: The Line of Best Fit
a) Use the data in Example 1 to find the equation of the line of best fit that relates the number of workers absent on an assembly line and the number of defective parts produced.
b) Graph the equation of the line of best fit on a scatter diagram that illustrates the set of bivariate points.

21. Example 3: The Line of Best Fit
Solution
From Example 1, we know that

22. Example 3: The Line of Best Fit
Solution
Now, find the y-intercept, b.

23. Example 3: The Line of Best Fit
Solution
The equation of the line of best fit is
y = mx + b
y = 3.23x

24. Example 3: The Line of Best Fit
Solution
To graph y = 3.23x , plot at least two points and draw the graph.
27.90 6
21.44 4
14.98 2 y x

25. Example 3: The Line of Best Fit
27.90 6
21.44 4
14.98 2 y x