1. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 12-1 Correlation and Regression

2. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-2 Correlation vs. Regression A scatter plot can be used to show the relationship between two variables Correlation analysis is used to measure the strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the relationship No causal effect is implied with correlation

3. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-3 Introduction to Regression Analysis Regression analysis is used to: Predict the value of a dependent variable based on the value of at least one independent variable Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to predict or explain Independent variable: the variable used to predict or explain the dependent variable

4. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-4 Simple Linear Regression Model Only one independent variable, X Relationship between X and Y is described by a linear function Changes in Y are assumed to be related to changes in X

5. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc. Chap 3-5 Scatter Plots of Sample Data with Various Coefficients of Correlation Y X Y X Y X Y X r = -1 r = -.6 r = +.3 r = +1 Y X r = 0

6. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-6 Types of Relationships Y X Y X Y Y X X Linear relationshipsCurvilinear relationships

7. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-7 Types of Relationships Y X Y X Y Y X X Strong relationshipsWeak relationships (continued)

8. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-8 Types of Relationships Y X Y X No relationship (continued)

9. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-9 Linear component Simple Linear Regression Model Population Y intercept Population Slope Coefficient Random Error term Dependent Variable Independent Variable Random Error component

10. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-10 (continued) Random Error for this X i value Y X Observed Value of Y for X i Predicted Value of Y for X i XiXi Slope = β 1 Intercept = β 0 εiεi Simple Linear Regression Model

11. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-11 The simple linear regression equation provides an estimate of the population regression line Simple Linear Regression Equation (Prediction Line) Estimate of the regression intercept Estimate of the regression slope Estimated (or predicted) Y value for observation i Value of X for observation i

12. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-12 b 0 is the estimated mean value of Y when the value of X is zero b 1 is the estimated change in the mean value of Y as a result of a one-unit change in X Interpretation of the Slope and the Intercept

13. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-13 Simple Linear Regression Example A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet) A random sample of 10 houses is selected Dependent variable (Y) = house price in $1000s Independent variable (X) = square feet

14. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-14 Simple Linear Regression Example: Data House Price in $1000s (Y) Square Feet (X) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700

15. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-15 Simple Linear Regression Example: Scatter Plot House price model: Scatter Plot

16. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-16 Simple Linear Regression Example: Using Excel

17. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-17 Simple Linear Regression Example: Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 The regression equation is:

18. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-18 Simple Linear Regression Example: Minitab Output The regression equation is Price = 98.2 + 0.110 Square Feet Predictor Coef SE Coef T P Constant 98.25 58.03 1.69 0.129 Square Feet 0.10977 0.03297 3.33 0.010 S = 41.3303 R-Sq = 58.1% R-Sq(adj) = 52.8% Analysis of Variance Source DF SS MS F P Regression 1 18935 18935 11.08 0.010 Residual Error 8 13666 1708 Total 9 32600 The regression equation is: house price = 98.24833 + 0.10977 (square feet)

19. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-19 Simple Linear Regression Example: Graphical Representation House price model: Scatter Plot and Prediction Line Slope = 0.10977 Intercept = 98.248

20. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-20 Simple Linear Regression Example: Interpretation of b o b 0 is the estimated mean value of Y when the value of X is zero (if X = 0 is in the range of observed X values) Because a house cannot have a square footage of 0, b 0 has no practical application

21. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-21 Simple Linear Regression Example: Interpreting b 1 b 1 estimates the change in the mean value of Y as a result of a one-unit increase in X Here, b 1 = 0.10977 tells us that the mean value of a house increases by 0.10977($1000) = $109.77, on average, for each additional one square foot of size

22. Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-22 Predict the price for a house with 2000 square feet: The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850 Simple Linear Regression Example: Making Predictions