2. Just one quick favor… Please use your phone or laptop Please take just a minute to complete Course Evaluations online….. Check your email for a link or go to… https://tce.oirps.arizona.edu/TCEOnline Lecture and Lab

3. Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Spring 2016 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays

5. Before our fourth and final exam (May 2 nd ) OpenStax Chapters 1 – 13 (Chapter 12 is emphasized) Plous Chapter 17: Social Influences Chapter 18: Group Judgments and Decisions Schedule of readings Study guide for Exam 4 is online Stats Review by Jonathon & Nick Wednesday evening (April 27 th ) Time: 6:30 – 8:30 Location: ILC 120 Cost: $5.00 Stats Review by Jonathon & Nick Wednesday evening (April 27 th ) Time: 6:30 – 8:30 Location: ILC 120 Cost: $5.00

6. On class website: Please complete homework worksheet #27 Completing 6 types of analysis using Excel: Semester Summary Due: Friday, April 29 th Homework

7. By the end of lecture today 4/27/16 Multiple Regression Using multiple predictor variables (independent) to make predictions about a single predicted variable (dependent) Multiple regression coefficients (b) One regression coefficient for each independent variable

8. No more labs this semester

9. Sample memorandum and general instructions for Project 4 are both online Due this week in class

10. Review (0.71 > 0.632) 50% is explained so the other 50% has yet to be explained

11. 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

12. Some useful terms 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) Coefficient of regression is name for “b” Residual is found by y – y’

13. Pop Quiz – 2.How many dependent variables are in a simple regression and in a multiple regression? 1.How does a multiple regression differ from a simple regression? (Give an example of each) 3. How are “slopes”, “b”s, and “regression coefficients” related? 4. Please name each symbol r r 2 b y – y’ 5. What possible values can each of these have? r r 2 b y – y’ Standard error of the estimate

14. Pop Quiz – 2.How many dependent variables are in a simple regression and in a multiple regression? 1.How does a multiple regression differ from a simple regression? (Give an example of each) 3. How are “slopes”, “b”s, and “regression coefficients” related? 4. Please name each symbol r r 2 b y – y’ 5. What possible values can each of these have? r r 2 b y – y’ Standard error of the estimate Simple regression has one predictor variable and one predicted variable Multiple regression has multiple predictor variables and one predicted variable Simple regression has one predictor variable and one predicted variable Multiple regression has multiple predictor variables and one predicted variable Examples: Simple regression: Predicting sales from number of sales calls made Multiple regression: Predicting job success from age, niceness, and harshness Examples: Simple regression: Predicting sales from number of sales calls made Multiple regression: Predicting job success from age, niceness, and harshness

15. Pop Quiz – 2.How many dependent variables are in a simple regression and in a multiple regression? 1.How does a multiple regression differ from a simple regression? (Give an example of each) 3. How are “slopes”, “b”s, and “regression coefficients” related? 4. Please name each symbol r r 2 b y – y’ 5. What possible values can each of these have? r r 2 b y – y’ Standard error of the estimate Simple regression has one independent variable and one dependent variable Multiple regression has multiple independent variables and one dependent variable Simple regression has one independent variable and one dependent variable Multiple regression has multiple independent variables and one dependent variable

16. Pop Quiz – 2.How many dependent variables are in a simple regression and in a multiple regression? 1.How does a multiple regression differ from a simple regression? (Give an example of each) 3. How are “slopes”, “b”s, and “regression coefficients” related? 4. Please name each symbol r r 2 b y – y’ 5. What possible values can each of these have? r r 2 b y – y’ Standard error of the estimate All are names for the same thing

17. Pop Quiz – 2.How many dependent variables are in a simple regression and in a multiple regression? 1.How does a multiple regression differ from a simple regression? (Give an example of each) 3. How are “slopes”, “b”s, and “regression coefficients” related? 4. Please name each symbol r r 2 b y – y’ 5. What possible values can each of these have? r r 2 b y – y’ Standard error of the estimate Coefficient of correlation Coefficient of determination Coefficient of regression Residual

18. Pop Quiz – 2.How many dependent variables are in a simple regression and in a multiple regression? 1.How does a multiple regression differ from a simple regression? (Give an example of each) 3. How are “slopes”, “b”s, and “regression coefficients” related? 4. Please name each symbol r r 2 b y – y’ 5. What possible values can each of these have? r r 2 b y – y’ Standard error of the estimate Can vary from -1 to +1 Can vary from 0 to +1 Any number Any positive number Any number

19. 14-18 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

26. 14-25 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.

27. 14-26 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.

28. 14-27 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.

30. 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?

32. 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

33. 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

34. + 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

35. + 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

36. + 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

37. + 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

38. + 427.19 - 4.5827 -14.8308 + 6.1010 427.19 - 4.5827 x 1 - 14.8308 x 2 + 6.1010 x 3 Y’ =

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. - 0.41107 No

46. + 1.2013 Yes - 0.41107 No

47. 0.0016 No + 1.2013 Yes - 0.41107 No

48. - 0.0019 No + 1.2013 Yes - 0.41107 No 0.0016

49. - 0.0019 No + 1.2013 Yes - 0.41107 No High School GPA 0.0016

50. - 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

51. 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

52. 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

53. 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

55. 19831.93 8196.32 50387 266 804 289.06 8196.32 28.3549 289.0618 Yes – close enough

56. 19831.93 289.06 1.96 2.58 19831.93 ± (1.96) (289.06) 19831.93 ± 566.56 19265.37 19265 20397.56 20397 19086.16 19086 20577.70 20577 19831.93 ± (2.58) (289.06) 19831.93 ± 745.77

58. Number of doors 22-doors vs 4-doors Quasi Price of car (dollars) Ratio Between Two-tailed $23,807.14 $20,580.67 alpha = 0.05 3.9677 1.9629 802 p =.000079 The average price for 2-door cars was $23,807.14, while the average price for 4-door cars was $20,580.67. A t-test was conducted and found this to be a significant difference t(802) = 3.9677; p < 0.01

59. Size of the engine 3 4- versus 6- versus 8 cylinder engines Quasi Price of car (dollars) Ratio Between $17,862 $20,081 alpha = 0.05 345.3577 3.006964 2 p =.(17 zeroes, then)69755 The average prices were $17,862, $20,081 and $38.968 for the 4-, 6-, and 8-cylinder engines (respectively). An ANOVA was conducted and found this to be a significant difference F(2,801) = 345; p < 0.01 801 $38,968

60. 802.195 The relationship between mileage and car price was -0.14. This is a weak and not significant correlation r(802) = -0.14305;n.s. -0.1431

61. df = 802 b = -0.1725 r = -0.1431 a = 24765 lower higher -0.1725x + 24765 For each additional mile driven (as x goes up by 1), cost of the car goes down by 17.25 cents. The base cost for the car (before taking into account the mileage) is $24,765 (30,000)(-0.1725) + 24765 = $19,590 -0.1431 2 =.0204633 or 2.04% The proportion of total variance for price of car that was accounted for by miles was 2.04%

62. Y’ = cost of car (dollars) mileagesize of car 3145.75 -0.15243 Y’ = 3145.75 + (-0.15243)(mileage) + (4027.67)( car size) 4027.67 yes 15.243 cents $4,027.67