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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
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Topics Dependent and independent variables Coefficient of correlation
Least squares regression
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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
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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
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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
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Perfect Negative Correlation
10 9 8 7 6 5 4 3 2 1 Y X Perfect Negative Correlation
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Perfect Positive Correlation
10 9 8 7 6 5 4 3 2 1 Y X Perfect Positive Correlation
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10 9 8 7 6 5 4 3 2 1 Y X Zero Correlation
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Strong Positive Correlation
10 9 8 7 6 5 4 3 2 1 Y X Strong Positive Correlation
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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
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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
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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.
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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
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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
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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
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Correlation analysis Exercises page 382 2 2 2 2 3 2
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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
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scatter diagram represent the judgments of four people.
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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
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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
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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
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= 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
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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
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Correlation analysis Exercises page 390 2 2 2 2 3 2
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