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Exam 4 – Optional Times for Final Two options for completing Exam 4 Thursday (12/4/14) – The regularly scheduled time Tuesday (12/9/14) – The optional later time Must sign up to take Exam 4 on Tuesday (12/2) Only need to take one exam – these are two optional times Sign up today to take Exam 4 at later date: December 9 th No need to sign up if you are taking it at regular time (December 4 th )

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Today 12/2/14 Use this as your study guide Multiple Regression Review Exam 4 Review Teacher Evaluations

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MGMT 276: Statistical Inference in Management Fall, 2014 Green sheets

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Reminder Talking or whispering to your neighbor can be a problem for us – please consider writing short notes. A note on doodling

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Exam 4 – Optional Times for Final Two options for completing Exam 4 Thursday (12/4/14) – The regularly scheduled time Tuesday (12/9/14) – The optional later time Must sign up to take Exam 4 on Tuesday (12/2) Only need to take one exam – these are two optional times

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Before our next exam (December 4 th ) Lind (10 – 12) Chapter 13: Linear Regression and Correlation Chapter 14: Multiple Regression Chapter 15: Chi-Square Plous (2, 3, & 4) Chapter 17: Social Influences Chapter 18: Group Judgments and Decisions Schedule of readings Study Guide online (shorter and longer version)

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Homework: No more homework!! My last name starts with a letter somewhere between A. A – D B. E – L C. M – R D. S – Z Please click in

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Multiple regression equations Can use variables to predict behavior of stock market probability of accident amount of pollution in a particular well quality of a wine for a particular year which candidates will make best workers

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Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a Measured current workers – the best workers tend to have highest “success scores”. (Success scores range from 1 – 1,000) Try to predict which applicants will have the highest success score. We have found that these variables predict success: Age (X 1 ) Niceness (X 2 ) Harshness (X 3 ) According to your research, age has only a small effect on success, while workers’ attitude has a big effect. Turns out, the best workers have high “niceness” scores and low “harshness” scores. Your results are summarized by this regression formula: Both 10 point scales Niceness (10 = really nice) Harshness (10 = really harsh) Success score = (1)( Age ) + (20)( Nice ) + (-75)( Harsh ) + 700 Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a Can use variables to predict which candidates will make best workers

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Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a According to your research, age has only a small effect on success, while workers’ attitude has a big effect. Turns out, the best workers have high “niceness” scores and low “harshness” scores. Your results are summarized by this regression formula: Success score = (1)( Age ) + (20)( Nice ) + (-75)( Harsh ) + 700

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Y’ is the dependent variable “Success score” is your dependent variable. X 1 X 2 and X 3 are the independent variables “Age”, “Niceness” and “Harshness” are the independent variables. Each “b” is called a regression coefficient. Each “b” shows the change in Y for each unit change in its own X (holding the other independent variables constant). a is the Y-intercept Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a According to your research, age has only a small effect on success, while workers’ attitude has a big effect. Turns out, the best workers have high “niceness” scores and low “harshness” scores. Your results are summarized by this regression formula: Success score = (1)( Age ) + (20)( Nice ) + (-75)( Harsh ) + 700

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14-13 The Multiple Regression Equation – Interpreting the Regression Coefficients b 1 = The regression coefficient for age (X 1 ) is “1” The coefficient is positive and suggests a positive correlation between age and success. As the age increases the success score increases. The numeric value of the regression coefficient provides more information. If age increases by 1 year and hold the other two independent variables constant, we can predict a 1 point increase in the success score. Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700

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14-14 The Multiple Regression Equation – Interpreting the Regression Coefficients b 2 = The regression coefficient for age (X 2 ) is “20” The coefficient is positive and suggests a positive correlation between niceness and success. As the niceness increases the success score increases. The numeric value of the regression coefficient provides more information. If the “niceness score” increases by one, and hold the other two independent variables constant, we can predict a 20 point increase in the success score. Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700

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14-15 The Multiple Regression Equation – Interpreting the Regression Coefficients b 3 = The regression coefficient for age (X 3 ) is “-75” The coefficient is negative and suggests a negative correlation between harshness and success. As the harshness increases the success score decreases. The numeric value of the regression coefficient provides more information. If the “harshness score” increases by one, and hold the other two independent variables constant, we can predict a 75 point decrease in the success score. Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700

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Here comes Victoria, her scores are as follows: Age = 30 Niceness = 8 Harshness = 2 What would we predict her “success index” to be? Y’ = = 3.812 Prediction line: Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a Y’ = 1X 1 + 20X 2 - 75X 3 + 700 Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 We predict Victoria will have a Success Index of 740 Y’ = 740 (1)(30) + (20)(8) - 75(2) + 700 Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700

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Here comes Victor, his scores are as follows: Here comes Victoria, her scores are as follows: Age = 30 Niceness = 8 Harshness = 2 What would we predict her “success index” to be? Y’ = = 3.812 We predict Victor will have a Success Index of 175 Prediction line: Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a Y’ = 1X 1 + 20X 2 - 75X 3 + 700 Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 Y’ = 740 (1)(30) + (20)(8) - 75(2) + 700 Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 Age = 35 Niceness = 2 Harshness = 8 We predict Victoria will have a Success Index of 740 What would we predict his “success index” to be? Y’ = Y’ = 175 (1)(35) + (20)(2) - 75(8) + 700 Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700

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We predict Victor will have a Success Index of 175 We predict Victoria will have a Success Index of 740 Can use variables to predict which candidates will make best workers Who will we hire?

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14-19 Can we predict heating cost? Three variables are thought to relate to the heating costs: (1) the mean daily outside temperature, (2) the number of inches of insulation in the attic, and (3) the age in years of the furnace. To investigate, Salisbury's research department selected a random sample of 20 recently sold homes. It determined the cost to heat each home last January Multiple Linear Regression - Example

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14-21 The Multiple Regression Equation – Interpreting the Regression Coefficients b 1 = The regression coefficient for mean outside temperature (X 1 ) is -4.583. The coefficient is negative and shows a negative correlation between heating cost and temperature. As the outside temperature increases, the cost to heat the home decreases. The numeric value of the regression coefficient provides more information. If we increase temperature by 1 degree and hold the other two independent variables constant, we can estimate a decrease of $4.583 in monthly heating cost.

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14-22 The Multiple Regression Equation – Interpreting the Regression Coefficients b 2 = The regression coefficient for mean attic insulation (X 2 ) is -14.831. The coefficient is negative and shows a negative correlation between heating cost and insulation. The more insulation in the attic, the less the cost to heat the home. So the negative sign for this coefficient is logical. For each additional inch of insulation, we expect the cost to heat the home to decline $14.83 per month, regardless of the outside temperature or the age of the furnace.

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14-23 The Multiple Regression Equation – Interpreting the Regression Coefficients b 3 = The regression coefficient for mean attic insulation (X 3 ) is 6.101 The coefficient is positive and shows a negative correlation between heating cost and insulation. As the age of the furnace goes up, the cost to heat the home increases. Specifically, for each additional year older the furnace is, we expect the cost to increase $6.10 per month.

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Applying the Model for Estimation What is the estimated heating cost for a home if: the mean outside temperature is 30 degrees, there are 5 inches of insulation in the attic, and the furnace is 10 years old?

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Multiple regression equations Prediction line Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a Very often we want to select students or employees who have the highest probability of success in our school or company. Andy is an administrator at a paralegal program and he wants to predict the Grade Point Average (GPA) for the incoming class. He thinks these independent variables will be helpful in predicting GPA. High School GPA (X 1 ) SAT - Verbal (X 2 ) SAT - Mathematical (X 3 ) Andy completes a multiple regression analysis and comes up with this regression equation: Y’ = 1.2X 1 +.00163X 2 -.00194X 3 -.411 Y’ = 1.2 gpa +.00163 sat verb -.00194sat math -.411

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Here comes Victoria, her scores are as follows: High School GPA = 3.81 SAT Verbal = 500 SAT Mathematical = 600 What would we predict her GPA to be in the paralegal program? Y’ = 1.2 (3.81) +.00163 (500) -.00194 (600) -.411 Y’ = 4.572 +.815 - 1.164 -.411 Y’ = 1.2 gpa +.00163 sat verb -.00194sat math -.411 Predict Victor’s GPA, his scores are as follows: High School GPA = 2.63 SAT - Verbal = 469 SAT - Mathematical = 440 Y’ = 1.2 (2.63) +.00163 (469) -.00194 (440) -.411 Y’ = 3.156 +.76447 -.8536 -.411 = 3.812 Y’ = 1.2 gpa +.00163 sat verb -.00194 sat math -.411 We predict Victor will have a GPA of 2.656 = 2.66 Prediction line: Y’ = b 1 X 1 + b 2 X 2 + b 3 X 3 + a Y’ = 1.2X 1 +.00163X 2 -.00194X 3 -.411 We predict Victoria will have a GPA of 3.812

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500 400 300 200 100 0 20 40 60 80 Average Temperature Heating Cost r(18) = - 0.50 r(18) = - 0.811508835 500 400 300 200 100 0 20 40 60 80 Insulation Heating Cost r(18) = - 0.40 r(18) = - 0.257101335 500 400 300 200 100 0 20 40 60 80 Age of Furnace Heating Cost r(18) = + 0.60 r(18) = + 0.536727562

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500 400 300 200 100 0 20 40 60 80 Average Temperature Heating Cost r(18) = - 0.50 r(18) = - 0.811508835 500 400 300 200 100 0 20 40 60 80 Insulation Heating Cost r(18) = - 0.40 r(18) = - 0.257101335 500 400 300 200 100 0 20 40 60 80 Age of Furnace Heating Cost r(18) = + 0.60 r(18) = + 0.536727562

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+ 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

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+ 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

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+ 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

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+ 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

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+ 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

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4.58 14.83 6.10 427.19 - 4.5827(30) -14.8308 (5) +6.1010 (10) Y’ = 427.19 - 137.481 - 74.154 + 61.010 Y’ = = $ 276.56 Calculate the predicted heating cost using the new value for the age of the furnace Use the regression coefficient for the furnace ($6.10), to estimate the change

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4.58 14.83 6.10 427.19 - 4.5827(30) -14.8308 (5) +6.1010 (10) Y’ = 427.19 - 137.481 - 74.154 + 61.010 Y’ = = $ 276.56 $ 276.56 Calculate the predicted heating cost using the new value for the age of the furnace Use the regression coefficient for the furnace ($6.10), to estimate the change 427.19 - 4.5827(30) -14.8308 (5) +6.1010 (10) Y’ = 427.19 - 137.481 - 74.154 + 61.010 Y’ = = $ 276.56 427.19 - 4.5827(30) -14.8308 (5) +6.1010 (11) Y’ = 427.19 - 137.481 - 74.154 + 67.111 Y’ = = $ 282.66 These differ by only one year but heating cost changed by $6.10 282.66 – 276.56 = 6.10

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4.0 3.0 2.0 1.0 0 1 2 3 4 High School GPA GPA r(7) = 0.50 r(7) = + 0.911444123 0 200 300 400 500 600 SAT (Verbal) GPA r(7) = + 0.80 r(7) = + 0.616334867 SAT (Mathematical) GPA r(7) = + 0.80 r(7) = + 0.487295007 4.0 3.0 2.0 1.0 4.0 3.0 2.0 1.0 0 200 300 400 500 600

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4.0 3.0 2.0 1.0 0 1 2 3 4 High School GPA GPA r(7) = 0.50 r(7) = + 0.911444123 0 200 300 400 500 600 SAT (Verbal) GPA r(7) = + 0.80 r(7) = + 0.616334867 SAT (Mathematical) GPA r(7) = + 0.80 r(7) = + 0.487295007 4.0 3.0 2.0 1.0 4.0 3.0 2.0 1.0 0 200 300 400 500 600

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4.0 3.0 2.0 1.0 0 1 2 3 4 High School GPA GPA r(7) = 0.50 r(7) = + 0.911444123 0 200 300 400 500 600 SAT (Verbal) GPA r(7) = + 0.80 r(7) = + 0.616334867 SAT (Mathematical) GPA r(7) = + 0.80 r(7) = + 0.487295007 4.0 3.0 2.0 1.0 4.0 3.0 2.0 1.0 0 200 300 400 500 600

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4.0 3.0 2.0 1.0 0 1 2 3 4 High School GPA GPA r(7) = 0.50 r(7) = + 0.911444123 0 200 300 400 500 600 SAT (Verbal) GPA r(7) = + 0.80 r(7) = + 0.616334867 SAT (Mathematical) GPA r(7) = + 0.80 r(7) = + 0.487295007 4.0 3.0 2.0 1.0 4.0 3.0 2.0 1.0 0 200 300 400 500 600

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- 0.41107 No

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+ 1.2013 Yes - 0.41107 No

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0.0016 No + 1.2013 Yes - 0.41107 No

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- 0.0019 No + 1.2013 Yes - 0.41107 No 0.0016

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- 0.0019 No + 1.2013 Yes - 0.41107 No High School GPA 0.0016

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- 0.0019 No + 1.2013 Yes - 0.41107 No High School GPA - 0.0019 x 3 + 0.0016 x 2 + 1.2013 x 1 Y’ = - 0.41107 0.0016

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1.201.0016.0019 - 0.0019 (460) + 0.0016 (430) + 1.2013 (2.8) Y’ = - 0.411 - 0.0019 x 3 + 0.0016 x 2 + 1.2013 x 1 Y’ = - 0.41107 = 2.76 2.76

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1.201.0016 - 0.0019 (460) + 0.0016 (430) + 1.2013 (3.8) Y’ = - 0.411 - 0.0019 x 3 + 0.0016 x 2 + 1.2013 x 1 Y’ = - 0.41107 = 3.96 3.96.0019

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1.201.0016.0019 Yes, use the regression coefficient for the HS GPA (1.2), to estimate the change = 1.8016 3.96 2.76 3.96 - 2.76 = 1.2

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Today we will be reviewing for the test using clicker questions. Please note these will not appear on the class website

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Just one quick favor… Please take just a minute to fill these out…..

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I'll be sitting outside Thank you for a wonderful semester! and good luck with your studies See you at the final exam.

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