1. AGENDA MULTIPLE REGRESSION BASICS  Overall Model Test (F Test for Regression)  Test of Model Parameters  Test of β i = β i *  Coefficient of Multiple Determination (R 2 ) Formula  Confidence Interval CORRELATION BASICS VI.Hypothesis Test on Correlation

2. Multiple Regression Basics Y=b 0 + b 1 X 1 + b 2 X 2 +…b k X k  Where Y is the predicted value of Y, the value lying on the estimated regression surface. The terms b 0,…,k are the least squares estimates of the population regression parameters ß i

3. I. ANOVA Table for Regression Analysis Source of Variation Degrees of Freedom Sums of Squares Mean SquaresF Regression kSSRMSR = SSR / kMSR/ MSE Residual n-k-1SSEMSE=SSE/(n-k-1) Total n-1SST

4. H0:β 1 = 0No Relationship H1: β 1 ≠ 0Relationship t-calc = n = sample size t-critical: II. Test of Model Parameters

5. III. Test of β i = β i * H0:β 1 = β i * H1: β 1 ≠ β i * t-calc = n =sample size t-critical:

6. R 2 = or IV. Coefficient of Multiple Determination (R 2 ) Formula Adjusted R 2 =

7. V. Confidence Interval Range of numbers believed to include an unknown population parameter.

8. Multiple Regression Example  Deciding where to locate a new retail store is one of the most important decisions that a manger can make.  The director of Blockbuster Video plans to use a regression model to help select a location for a new store. She decides to use the annual gross revenue as a measure of success (Y). She uses a sample of 50 stores.

9. Determinants of Success (X 1 ) =Number of people living within one mile of the store (X 2 ) =Mean income of households within one mile of the store (X 3 ) = Number of Competitors within one mile of the store (X 4 ) =Rental price of a newly released movie

10. Output from Computer Regression Line: Y= -20297+6.44X 1 +7.27X 2 -6,709X 3 +15,969X 4

11. Multiple Regression Example Conduct the following tests: Overall Model F test Test whether β 2 = 0 (s b2 = 3.705) Test whether β 3 = -5000 (s b3 = 3,818) What is the R 2 ? the adjusted R 2 ? Construct a 95% confidence interval for β 4 (s b4 = 10,219)

12. Correlation  Measures the strength of the linear relationship between two variables  Ranges from -1 to 1  Positive = direct relationship  Negative = inverse relationship  Near 0 = no strong linear relationship  Does NOT imply causality

13. Illustrations of correlation Y X r=0 Y X r=-.8 Y X r=.8 Y X r=0 Y X r=-1 Y X r=1

14. VI. Hypothesis Test on Correlation  To test the significance of the linear relationship between two random variables: H 0 :  = 0 no linear relationship H 1 :   0 linear relationship  This is a t-test with (n-2) degrees of freedom:

15. VI. Hypothesis Test on Correlation (cont.)  Is the number of penalty flags thrown by Big Ten Officials linearly related to the number of points scored by the football team? (n=100) Sxy= - 59 Sx= 7.45 Sy= 9.10