Simple linear 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. Dr. Ahmed M. Sultan1

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 Dr. Ahmed M. Sultan2

Airline Cost Data Number of Passengers X Cost ($1,000) Y Dr. Ahmed M. Sultan3

Scatter Plot of Airline Cost Data Dr. Ahmed M. Sultan4

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Regression Models  Deterministic Regression Model Y =  0 +  1 X  Probabilistic Regression Model Y =  0 +  1 X +    0 and  1 are population parameters   0 and  1 are estimated by sample statistics b 0 and b 1

Equation of the Simple Regression Line

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

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

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Solving for b 1 and b 0 of the Regression Line: Airline Cost Example (Part 1)

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Solving for b 1 and b 0 of the Regression Line: Airline Cost Example (Part 2)

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

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Airline Cost: Excel Summary Output SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations12 ANOVA dfSSMSFSignificance F Regression E-06 Residual Total CoefficientsStandard Errort StatP-value Intercept Number of Passengers E-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 Excel Graph of Residuals for the Airline Cost Example

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Nonlinear Residual Plot 0 X

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Nonconstant Error Variance 0 X 0 X

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Graphs of Nonindependent Error Terms 0 X 0 X

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Healthy Residual Plot 0 X

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

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

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

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

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

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

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

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

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Testing the Overall Model (Part 2) ANOVA dfSSMSFSignificance F Regression E-06 Residual Total F = > 4.96, reject H 0

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 Confidence Interval to Estimate  Y : Airline Cost Example

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

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Confidence Intervals for Estimation Number of Passengers C o s t Regression 95% CI 95% PI Regression Plot

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

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

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Three Degrees of Correlation r < 0r > 0 r = 0

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