2. Lecturer’s desk Physics- atmospheric Sciences (PAS) - Room 201 s c r e e n Row A Row B Row C Row D Row E Row F Row G Row H 131211109 87 Row A 14131211109 87 Row B 1514131211109 87 Row C 1514131211109 87 Row D 16 1514131211109 87 Row E 17 16 1514131211109 87 Row F 1716 1514131211109 87 Row G 1716 1514131211109 87 Row H 16 18 table Row A Row B Row C Row D Row E Row F Row G Row H 15141716 1819 16 15 18171920 17161918 2021 18172019 2122 19182120 2223 20192221 2324 18172019 2122 19182120 2223 2143 56 2143 56 2143 56 2143 56 2143 56 2143 56 2143 56 2143 56 Row J Row K Row L Row M Row N Row P 2143 5 2143 5 2143 5 2143 5 2143 5 1 5 Row J Row K Row L Row M Row N Row P 27262928 30 25242726 28 24232625 27 23222524 26 25242726 28 27262928 30 6 14 131211109 87 16151817 19 202122 614131211109 87 16 15 18 17 19 20212223 614131211109 87 16 15 18171920 2122 23 6 14 131211109 87 1624181719 20 2122 231525 6 14 131211109 87 1624181719 20 2122 231525 Row Q 2143 5 27262928 30 6 14 131211109 87 242223 21 - 15 25 37363938 40 34 3132 3335 69 87 13 table 14 18 192021

3. MGMT 276: Statistical Inference in Management Fall 2015

5. Before our next exam (December 3 rd ) OpenStax Chapters 1 – 13 (Chapter 12 is emphasized) Plous Chapter 17: Social Influences Chapter 18: Group Judgments and Decisions Schedule of readings Stats Review by Jonathon & Nick Wednesday evening (December 2 nd ) Time: 6:30 – 8:30 Location: ILC 120 Cost: $5.00 Stats Review by Jonathon & Nick Wednesday evening (December 2 nd ) Time: 6:30 – 8:30 Location: ILC 120 Cost: $5.00

6. Logic of hypothesis testing with Correlations Interpreting the Correlations and scatterplots Simple and Multiple Regression Using correlation for predictions r versus r 2 Regression uses the predictor variable (independent) to make predictions about the predicted variable (dependent) Coefficient of correlation is name for “r” Coefficient of determination is name for “r 2 ” (remember it is always positive – no direction info) Standard error of the estimate is our measure of the variability of the dots around the regression line (average deviation of each data point from the regression line – like standard deviation) Coefficient of regression will “b” for each variable (like slope) Over next couple of lectures 11/19/15

7. On class website: Please print and complete homework worksheet #17, 18 and 19 Summarizing seven prototypical designs (note this is worth three homework assignments) Homework due – Tuesday (December 1 st )

8. Regression Example Rory is an owner of a small software company and employs 10 sales staff. Rory send his staff all over the world consulting, selling and setting up his system. He wants to evaluate his staff in terms of who are the most (and least) productive sales people and also whether more sales calls actually result in more systems being sold. So, he simply measures the number of sales calls made by each sales person and how many systems they successfully sold.

9. Review

10. Summary Slope: as sales calls increase by one, 11.579 more systems should be sold Intercept: suggests that we can assume each salesperson will sell at least 20.526 systems Review

11. Pop Quiz - 5 Questions 2. What is a residual? How would you find it? 1. What is regression used for? Include and example 3. What is Standard Error of the Estimate (How is it related to residuals?) 4. Give one fact about r 2 5. How is regression line like a mean?

12. Pop Quiz - 5 Questions Regressions are used to take advantage of relationships between variables described in correlations. We choose a value on the independent variable (on x axis) to predict values for the dependent variable (on y axis). 1. What is regression used for? Include and example

13. Writing Assignment - 5 Questions 2. What is a residual? How would you find it? Residuals are the difference between our predicted y (y’) and the actual y data points. Once we choose a value on our independent variable and predict a value for our dependent variable, we look to see how close our prediction was. We are measuring how “wrong” we were, or the amount of “error” for that guess. Y – Y’

14. Writing Assignment - 5 Questions 3. What is Standard Error of the Estimate (How is it related to residuals?) The average length of the residuals The average error of our guess The average length of the green lines The standard deviation of the regression line

15. Writing Assignment - 5 Questions 4. Give one fact about r 2 5. How is regression line like a mean?

17. 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 Review

18. 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 Review

19. 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 Review

20. 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 Review

21. 14-20 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 Review

22. 14-21 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 Review

23. 14-22 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

24. 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

25. 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

26. 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?

27. Conducting multiple regression analyses that are relevant and useful starts with measurement designed to decrease uncertainty “Anything can be measured. If a thing can be observed in any way at all, it lends itself to some type of measurement method. No matter how “fuzzy” the measurement is, it’s still a measurement if it tells you more than you knew before.” Douglas Hubbard -Author “How to Measure Anything: Finding the value of “Intangibles” in Business”

28. Measurements don’t have to be precise to be useful “Anything can be measured. If a thing can be observed in any way at all, it lends itself to some type of measurement method. No matter how “fuzzy” the measurement is, it’s still a measurement if it tells you more than you knew before.” Douglas Hubbard -Author “How to Measure Anything: Finding the value of “Intangibles” in Business” How do we operationally define and measure constructs that we care about? “A problem well stated is a problem half solved” Charles Kettering (1876 – 1958), American inventor, holder of 300 patents, including electrical ignition for automobiles “It is better to be approximately right, than to be precisely wrong.” - Warren Buffett

29. 14-28 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

31. 14-30 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.

32. 14-31 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.

33. 14-32 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.

40. 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?

41. 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

42. 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

44. 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

45. 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

46. + 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

47. + 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

48. + 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

49. + 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

50. + 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

51. 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

52. 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

53. 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

54. 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

55. 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

56. 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

57. - 0.41107 No

58. + 1.2013 Yes - 0.41107 No

59. 0.0016 No + 1.2013 Yes - 0.41107 No

60. - 0.0019 No + 1.2013 Yes - 0.41107 No 0.0016

61. - 0.0019 No + 1.2013 Yes - 0.41107 No High School GPA 0.0016

62. - 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

63. 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

64. 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

65. 1.201.0016.0019 Yes, use the regression coefficient for the HS GPA (1.2), to estimate the change 3.96 2.76 3.96 - 2.76 = 1.2