Business Statistics, 4e by Ken Black Chapter 13 Simple Regression Analysis Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Compute the equation of a simple regression line from a sample of data, and interpret the slope and intercept of the equation. Understand the usefulness of residual analysis in testing the assumptions underlying regression analysis and in examining the fit of the regression line to the data. Compute a standard error of the estimate and interpret its meaning. Compute a coefficient of determination and interpret it. Test hypotheses about the slope of the regression model and interpret the results. Estimate values of Y using the regression model. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

Regression and Correlation Regression analysis is the process of constructing a mathematical model or function that can be used to predict or determine one variable by another variable. Correlation is a measure of the degree of relatedness of two variables. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Simple Regression Analysis bivariate (two variables) linear regression -- the most elementary regression model dependent variable, the variable to be predicted, usually called Y independent variable, the predictor or explanatory variable, usually called X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Airline Cost Data Number of Passengers X Cost ($1,000) Y 61 4.280 63 4.080 67 4.420 69 4.170 70 4.480 74 4.300 76 4.820 81 4.700 86 5.110 91 5.130 95 5.640 97 5.560 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

Scatter Plot of Airline Cost Data Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Regression Models Deterministic Regression Model Y = 0 + 1X Probabilistic Regression Model Y = 0 + 1X +  0 and 1 are population parameters 0 and 1 are estimated by sample statistics b0 and b1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

Equation of the Simple Regression Line Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

Least Squares Analysis Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

Least Squares Analysis Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Solving for b1 and b0 of the Regression Line: Airline Cost Example (Part 1) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

Solving for b1 and b0 of the Regression Line: Airline Cost Example (Part 2) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Graph of Regression Line for the Airline Cost Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

Airline Cost: Excel Summary Output Regression Statistics Multiple R 0.94820033 R Square 0.89908386 Adjusted R Square 0.88899225 Standard Error 0.17721746 Observations 12 ANOVA   df SS MS F Significance F Regression 1 2.79803 89.092179 2.7E-06 Residual 10 0.31406 0.03141 Total 11 3.11209 Coefficients t Stat P-value Intercept 1.56979278 0.33808 4.64322 0.0009175 Number of Passengers 0.0407016 0.00431 9.43887 2.692E-06 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Residual Analysis: Airline Cost Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

Excel Graph of Residuals for the Airline Cost Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

Nonlinear Residual Plot X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

Nonconstant Error Variance X X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

Graphs of Nonindependent Error Terms X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Healthy Residual Plot X 21 X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

Standard Error of the Estimate Sum of Squares Error Standard Error of the Estimate Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

Determining SSE for the Airline Cost Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23

Standard Error of the Estimate for the Airline Cost Example Sum of Squares Error Standard Error of the Estimate Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 24

Coefficient of Determination Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Coefficient of Determination for the Airline Cost Example 89.9% of the variability of the cost of flying a Boeing 737 is accounted for by the number of passengers. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Hypothesis Tests for the Slope of the Regression Model Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Hypothesis Test: Airline Cost Example (Part 1) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Hypothesis Test: Airline Cost Example (Part 2) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Testing the Overall Model (Part 1) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Testing the Overall Model (Part 2) ANOVA   df SS MS F Significance F Regression 1 2.79803 89.092179 2.7E-06 Residual 10 0.31406 0.03141 Total 11 3.11209 F = 89.09 > 4.96, reject H0 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Point Estimation for the Airline Cost Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 29

Confidence Interval to Estimate Y : Airline Cost Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 30

Confidence Interval to Estimate the Average Value of Y for some Values of X: Airline Cost Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 31

Prediction Interval to Estimate Y for a given value of X Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 32

Confidence Intervals for Estimation 6 7 8 9 1 4 5 N u m b e r o f P a s n g C t R i % I l Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 34

MINITAB Regression Analysis of the Airline Cost Example Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 35

Pearson Product-Moment Correlation Coefficient Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 39

Three Degrees of Correlation Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 40