1. Chapter 13 Linear Regression and Correlation Basic Statistics
for Business and Economics
Fifth Edition
Chapter 13
Linear Regression and Correlation
Douglas William Samuel
Irwin/McGraw-Hill 1 1 1 2 1 1

2. Topics Dependent and independent variables Coefficient of correlation
Least squares regression

3. Scatter diagram
Describing the relationship between two variables; There are situations where we have to study a relationship between two variables, this called bivariate data. Examples;
- A relationship between stocks and the real state industry
- A relationship between selling vehicles and the age of buyer.
Scatter diagram; The graphical technique we use to show the relationship between two variables (at least interval or ratio levels). Figure 1 2 2 2 2 3 2

4. Scatter diagram Vehicle Price Age
The Scatter diagram shows a positive relationship between variables. As
Age increased the vehicle selling price also increased. In fact
Older buyers tend to buy more expensive cars. 2 2 2 3 2 2

5. Correlation analysis
Correlation analysis is a study of the relationship between variables. The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables. It can range from to Values of or 1.00 indicate perfect and strong correlation. Negative values indicate an inverse relationship and positive values indicate a direct relationship. Values close to 0.0 indicate weak correlation 2 2 2 2 3 2

6. Perfect Negative Correlation10 9 8 7 6 5 4 3 2 1 Y X
Perfect Negative Correlation

7. Perfect Positive Correlation10 9 8 7 6 5 4 3 2 1 Y X
Perfect Positive Correlation

8. 10 9 8 7 6 5 4 3 2 1 Y X
Zero Correlation

9. Strong Positive Correlation10 9 8 7 6 5 4 3 2 1 Y X
Strong Positive Correlation

10. Correlation analysis
Example; Suppose the sales manager of Copier Sales of America wants to determine whether there is a relationship between the number of sales calls made in a month and the number of copiers sold that month. 2 2 2 2 3 2

11. Sales rep # sales calls # copier sold Tom 20 30 Jeff 40 60 Brian 20 40
Independent variable
dependent variable X Y
Sales rep # sales calls # copier sold
Tom
Jeff
Brian
Greg
Susan
Carlos
Rich
Mike
Mark
Soni 2 2 2 3 2 2

12. Scatter diagram showing sales calls and copiers sold
Scatter diagram showing sales calls and copiers sold. The diagram shows as the calls increases as the selling of copiers increases.

13. Sales rep # sales calls # copier sold Tom 20 30
Jeff sales mean =
Brian /10=22
Greg copiers mean =
Susan /10 =45
Carlos
Rich
Mike
Mark
Soni 2 2 2 2 3 2

14. Sales rep (x-x) y-y (x-x) (y-y) Tom 20-22= -2 30-45 = -15 30
Jeff = =
Brian = =
Greg = =
Susan = =
Carlos = =
Rich = =
Mike = =
Mark = =
Soni = =
900 2 2 2 2 3 2

15. Correlation analysis Correlation coefficient r =
Sx is the standard deviation of sales calls= 9.189
Sy is the standard deviation of copier sold =
r = .759
The coefficient determination = (r)² = (.759)²
= .576
S(X – X)(Y – Y)
(n-1)sxsy
r =
(10-1)(9.189(14.337) 2 2 2 3 2 2

16. Correlation analysis
Exercises page 382 2 2 2 2 3 2

17. Regression analysis
Technique used to develop the equation to express the linear ( straight line) relationship between two variables. The equation will estimate the value of the dependent variable (Y) based on the selected value of the independent variable (X). If we take our previous example about the sales people calls, the following scatter diagram represent the judgments of four people; 2 2 2 2 3 2

18. scatter diagram represent the judgments of four people.

19. Regression analysis
Judgment will be eliminated by determining the regression line using a mathematical method called the “ Least squares principle”
There are three equation to draw the line;
(1) Y΄ = a + bX
Y΄; read, Y prime, is the predicted value of the
Y variable for a selected X value.
a; is the Y intercept, it’s the estimated of Y when X=0
b; is the Slope of the line.
X; is any value of the independent variable that is selected. 2 2 2 2 3 2

20. Regression analysis (2) b = r Sy Sx r ; is the correlation coefficient
Sy; is the standard deviation of Y
Sx; is the standard deviation of X 2 2 2 2 3 2

21. Regression analysis (3) a= Y - bX Y; is the mean of Y values
X; is the mean of X values 2 2 2 2 3 2

22. = 42.6316 copiers to be sold in 20 calls
Example; recall the example involving Copier Sales of America. What is the expected number of copiers sold by a representative who made 20 calls ?
Steps of the solution:
(1) b = r Sy Sx
b = .759 ( ) =
9.189
(2) a= Y - bX
a = 45 – (1.1842) 22 =
(3) Y΄ = a + bX
= X X = 20 calls
= copiers to be sold in 20 calls 2 2 2 2 3 2

23. The Line of regression Drawn on the Scatter Diagram
You need to know two points to draw the line: First : Y = when X =0 Second: Y = X X = highest calls Y = (40) = Y
X= 40
Y =
X= 0
Y = X
The Line of regression Drawn on the Scatter Diagram

24. Correlation analysis
Exercises page 390 2 2 2 2 3 2