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AGENDA

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 Hypothesis Test on Correlation. Multiple Regression Basics.

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AGENDA

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  1. AGENDA MULTIPLE REGRESSION BASICS • Overall Model Test (F Test for Regression) • Test of Model Parameters • Test of βi = βi* • Coefficient of Multiple Determination (R2) Formula • Confidence Interval CORRELATION BASICS • Hypothesis Test on Correlation

  2. Multiple Regression Basics Y=b0 + b1X1 + b2X2 +…bkXk • Where Y is the predicted value of Y, the value lying on the estimated regression surface. The terms b0,…,k are the least squares estimates of the population regression parameters ßi

  3. I. ANOVA Table for Regression Analysis

  4. II. Test of Model Parameters H0: β1= 0 No Relationship H1: β1 ≠ 0 Relationship t-calc = n = sample size t-critical:

  5. III. Test of βi = βi* H0: β1= βi* H1: β1≠ βi* t-calc = n = sample size t-critical:

  6. IV. Coefficient of Multiple Determination (R2) Formula R2 = or Adjusted R2 =

  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 (X1) = Number of people living within one mile of the store (X2) = Mean income of households within one mile of the store (X3) = Number of Competitors within one mile of the store (X4) = Rental price of a newly released movie

  10. Output from Computer Regression Line: Y= -20297+6.44X1+7.27X2-6,709X3+15,969X4

  11. Multiple Regression Example Conduct the following tests: • Overall Model F test • Test whether β2 = 0 (sb2 = 3.705) • Test whether β3 = -5000 (sb3 = 3,818) • What is the R2? the adjusted R2? • Construct a 95% confidence interval for β4 (sb4 = 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. Y Y r=1 r=-1 X X Y Y Y r=-.8 r=0 r=.8 X X X Illustrations of correlation Y r=0 X

  14. VI. Hypothesis Test on Correlation • To test the significance of the linear relationship between two random variables: H0: =0 no linear relationship H1: 0linear 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

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