2. Stage Screen Row B 13 121110 20191817 14 13 121110 19181716 1514 Gallagher Theater 16 65879 Row R 6 58 7 9 Lecturer’s desk Row A Row B Row C 4 3 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 3 21 3 2 43 21 Row A 17 16 15 Row A Row C 131211 10 1514 6 58 7 9 Row D 13121110 1514 16 6 58 7 9 20191817 Row D Row E 131211 10 1514 6 58 7 9 19181716 Row E Row F 13121110 1514 16 6 58 7 9 20191817 Row F Row G 13121110 1514 6 58 7 9 19181716 Row G Row H 13121110 1514 16 6 58 7 9 20191817 Row H Row I 13121110 1514 6 58 7 9 19181716 Row I Row J 13121110 1514 16 6 58 7 9 20191817 Row J Row K 13121110 1514 6 58 7 9 19181716 Row K Row L 13121110 1514 16 6 58 7 9 20 191817 Row L Row M 13121110 1514 6 58 7 9 19181716 Row M Row N 13121110 1514 16 6 5879 20191817 Row N Row O 13121110 1514 6 58 7 9 19181716 Row O Row P 13121110 1514 16 6 5879 20191817 Row P Row Q 13121110 6 5879 161514 Row Q 4 4 Row R 10 879 Row S Row B Row C Row D Row E Row F Row G Row H Row I Row J Row K Row L Row M Row N Row O Row P Row Q 26Left-Handed Desks A14, B16, B20, C19, D16, D20, E15, E19, F16, F20, G19, H16, H20, I15, J16, J20, K19, L16, L20, M15, M19, N16, P20, Q13, Q16, S4 5 Broken Desks B9, E12, G9, H3, M17 Need Labels B5, E1, I16, J17, K8, M4, O1, P16 Left handed

3. Stage Screen 2213 121110 Row A Row B Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 17 Row C Row D Row E Projection Booth 65 4 table Row C Row D Row E 30 27 26252423 282726 2524 23 3127262524 23 R/L handed broken desk 16 1514 13 12 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 Social Sciences 100 Row N Row O Row P Row Q Row R 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 8 7 9 65 4 8 7 9 3 2 6 5 48793 2 1 6 5 48793 2 1 Row F Row G Row H Row J Row K Row L Row M Row N Row O Row P Row Q Row R 6 5 48793 2 1 6 5 48793 2 1 Row I 2213 121110 2019181716151421 Row I 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 Lecturer’s desk 6 5 48793 2 1 262524 23 302928 Row F Row G Row H Row J Row K Row L Row M Row N Row O Row P Row Q Row R Row I 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 Row B 2928 27

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

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

7. 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

8. 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

9. Homework due – Tuesday (November 25 th ) On class website: Please print and complete homework worksheet #18 Hypothesis Testing with correlation coefficients and simple regression

10. Next couple of lectures 11/20/14 Use this as your study guide 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)

11. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? ( α =.05 or.01)? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed r is bigger than critical r then reject null Step 5: Conclusion - tie findings back in to research problem Critical statistic (e.g. critical r) value from table? For correlation null is that r = 0 (no relationship) Degrees of Freedom = (n – 2) df = # pairs - 2 Review

12. Finding a statistically significant correlation The result is “statistically significant” if: the observed correlation is larger than the critical correlation we want our r to be big if we want it to be significantly different from zero!! (either negative or positive but just far away from zero) the p value is less than 0.05 (which is our alpha) we want our “p” to be small!! we reject the null hypothesis then we have support for our alternative hypothesis Review

13. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? ( α =.05 or.01)? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed r is bigger than critical r then reject null Step 5: Conclusion - tie findings back in to research problem Critical statistic (e.g. critical r) value from table? For correlation null is that r = 0 (no relationship) Degrees of Freedom = (n – 2) df = # pairs - 2 Review

14. Five steps to hypothesis testing Problem 1 Is there a relationship between the: Price Square Feet We measured 150 homes recently sold

15. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule – find critical r (from table) Alpha level? ( α =.05) null is that there is no relationship (r = 0.0) Degrees of Freedom = (n – 2) df = # pairs - 2 Is there a relationship between the cost of a home and the size of the home alternative is that there is a relationship (r ≠ 0.0) 150 pairs – 2 = 148 pairs

16. Critical r value from table df = # pairs - 2 df = 148 pairs α =.05 Critical value r (148) = 0.195

17. Five steps to hypothesis testing Step 3: Calculations

18. Five steps to hypothesis testing Step 3: Calculations

19. Five steps to hypothesis testing Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed r is bigger than critical r then reject null r = 0.726965 Critical value r (148) = 0.195 Observed correlation r (148) = 0.726965 Yes we reject the null 0.727 > 0.195

20. Conclusion: Yes we reject the null. The observed r is bigger than critical r (0.727 > 0.195) Yes, this is significantly different than zero – something going on These data suggest a strong positive correlation between home prices and home size. This correlation was large enough to reach significance, r(148) = 0.73; p < 0.05

21. Finding a statistically significant correlation The result is “statistically significant” if: the observed correlation is larger than the critical correlation we want our r to be big if we want it to be significantly different from zero!! (either negative or positive but just far away from zero) the p value is less than 0.05 (which is our alpha) we want our “p” to be small!! we reject the null hypothesis then we have support for our alternative hypothesis

22. Correlation matrices Correlation matrix: Table showing correlations for all possible pairs of variables 1.0** EducationAgeIQIncome IQ Age Education Income 1.0** 0.65** 0.52* 0.27* 0.41* 0.38* -0.02 * p < 0.05 ** p < 0.01 Remember, Correlation = “r”

23. Correlation matrices Correlation matrix: Table showing correlations for all possible pairs of variables EducationAgeIQIncome IQ Age Education Income 0.65** 0.52* 0.27* 0.41*0.38* -0.02 * p < 0.05 ** p < 0.01

24. Finding a statistically significant correlation The result is “statistically significant” if: the observed correlation is larger than the critical correlation we want our r to be big if we want it to be significantly different from zero!! (either negative or positive but just far away from zero) the p value is less than 0.05 (which is our alpha) we want our “p” to be small!! we reject the null hypothesis then we have support for our alternative hypothesis

25. Variable names Make up any name that means something to you VARX = “Variable X” VARY = “Variable Y” VARZ = “Variable Z” Correlation of X with X Correlation of Y with Y Correlation of Z with Z Correlation matrices

26. Variable names Make up any name that means something to you VARX = “Variable X” VARY = “Variable Y” VARZ = “Variable Z” Correlation of X with Y Correlation matrices p value for correlation of X with Y p value for correlation of X with Y Does this correlation reach statistical significance?

27. Variable names Make up any name that means something to you VARX = “Variable X” VARY = “Variable Y” VARZ = “Variable Z” Correlation of X with Z p value for correlation of X with Z p value for correlation of X with Z Correlation matrices Does this correlation reach statistical significance?

28. Variable names Make up any name that means something to you VARX = “Variable X” VARY = “Variable Y” VARZ = “Variable Z” Correlation of Y with Z p value for correlation of Y with Z p value for correlation of Y with Z Correlation matrices Does this correlation reach statistical significance?

29. What do we care about? Correlation matrices

30. What do we care about? We measured the following characteristics of 150 homes recently sold Price Square Feet Number of Bathrooms Lot Size Median Income of Buyers

31. Correlation matrices What do we care about?

32. Correlation matrices What do we care about?

33. Correlation matrices What do we care about?

34. Correlation matrices What do we care about? Review Critical value r (148) = 0.195

35. Correlation: Independent and dependent variables When used for prediction we refer to the predicted variable as the dependent variable and the predictor variable as the independent variable Dependent Variable Dependent Variable Independent Variable Independent Variable What are we predicting?

36. Correlation Positive correlation: as values on one variable go up, so do values for the other variable Negative correlation: as values on one variable go up, the values for the other variable go down Yearly income by expenses per year Positive Correlation Expenses per year Yearly Income What are we predicting?

37. Correlation Positive correlation: as values on one variable go up, so do values for the other variable Negative correlation: as values on one variable go up, the values for the other variable go down Temperatures by time spent outside in Tucson in summer Negative Correlation Time outside Temperature What are we predicting?

38. Correlation Positive correlation: as values on one variable go up, so do values for the other variable Negative correlation: as values on one variable go up, the values for the other variable go down Height by average driving speed Zero Correlation Average Speed Height What are we predicting?

39. Correlation Positive correlation: as values on one variable go up, so do values for the other variable Negative correlation: as values on one variable go up, the values for the other variable go down Amount Healthtex spends per month on advertising by sales in the month Positive Correlation Amount spent On Advertising Amount of sales What are we predicting?

40. Correlation - What do we need to define a line Expenses per year Yearly Income Y-intercept = “a” ( also “b 0 ”) Where the line crosses the Y axis Slope = “b” ( also “b 1 ”) How steep the line is If you spend this much If you probably make this much The predicted variable goes on the “Y” axis and is called the dependent variable The predictor variable goes on the “X” axis and is called the independent variable

41. Angelina Jolie Buys Brad Pitt a $24 million Heart-Shaped Island for his 50th Birthday Expenses per year Yearly Income Angelina spent this much Angelina probably makes this much Dustin spends $12 for his Birthday Dustin spent this much Dustin probably makes this much Revisit this slide

42. Assumptions Underlying Linear Regression These Y values are normally distributed. The means of these normal distributions of Y values all lie on the straight line of regression. For each value of X, there is a group of Y values The standard deviations of these normal distributions are equal.

43. Correlation - the prediction line Prediction line makes the relationship easier to see (even if specific observations - dots - are removed) identifies the center of the cluster of (paired) observations identifies the central tendency of the relationship (kind of like a mean) can be used for prediction should be drawn to provide a “best fit” for the data should be drawn to provide maximum predictive power for the data should be drawn to provide minimum predictive error - what is it good for?

44. Expenses per year Yearly Income Expenses per year Yearly Income Expenses per year Yearly Income Correlation - What do we need to define a line Y-intercept = “a” Where the line crosses the Y axis Slope = “b” How steep the line is Y-intercept is good… slope is wrong Y-intercept is wrong… slope is good

45. Number Cavities Brushing Teeth Number Cavities Brushing Teeth Number Cavities Brushing Teeth Correlation - What do we need to define a line Y-intercept is good… slope is wrong Y-intercept is wrong… slope is good Y-intercept = “a” Where the line crosses the Y axis Slope = “b” How steep the line is

47. Notice in this case it is negative Interpreting regression equation a) Interpret the slope of the fitted regression line: Sales = 842 – 37.5 Price A slope of “37.5” suggests that raising “price” by 1 unit will reduce “sales” by 37.5 units b) If “price” = 20, what is the prediction for “Sales”? Sales = 842 – 37.5 Price Sales = 842 - 37.5 Price Sales = 842 - (37.5) (20) Sales = 842 - (37.5) (20) = 842 – 750 = 92 price of product Sales Prediction line Y’ = a + b 1 X 1 Y’ = 842 + (-37.5)X 1 Y-intercept Slope

48. a) Interpret the slope of the fitted regression line: Sales = 842 – 37.5 Price A slope of “37.5” suggests that raising “price” by 1 unit will reduce “sales” by 37.5 units b) If “price” = 20, what is the prediction for “Sales”? Sales = 842 – 37.5 Price Sales = 842 - 37.5 Price Sales = 842 - (37.5) (20) Sales = 842 - (37.5) (20) = 842 – 750 = 92 (20, 92) price of product Sales Prediction line Y’ = a + b 1 X 1 Y’ = 842 + (-37.5)X 1 Y-intercept Slope If Price = 20 Sales probably about 92 units Interpreting regression equation

49. a) The regression equation: NetIncome = 2,277 +.0307 Revenue Interpret the slope Prediction line Y’ = a + b 1 X 1 Y’ = 2.277 + (.0307)X 1 A slope of “.0307” suggests that raising “Revenue” by 1 dollar, NetIncome will raise by 3 cents b) If “Revenue” = 1,000, what is the prediction for “NetIncome”? NetIncome = 2,277 +.0307 Revenue NetIncome = 2,277 + (.0307 )(1,000) NetIncome = 2,277 + 30.7 = 2,307.7 (1,000, 2,307.7) Revenue NetIncome Notice in this case it is positive Y-intercept Slope Interpreting regression equation

50. a) The regression equation: NetIncome = 2,277 +.0307 Revenue Interpret the slope Prediction line Y’ = a + b 1 X 1 Y’ = 2,277 + (.0307)X 1 Y-intercept Slope A slope of “.0307” suggests that raising “Revenue” by 1 dollar, NetIncome will raise by 3 cents b) If “Revenue” = 1,000, what is the prediction for “NetIncome”? NetIncome = 2,277 +.0307 Revenue NetIncome = 2,277 + (.0307 )(1,000) NetIncome = 2,277 + 30.7 = 2,307.7 (1,000, 2,307.7) Revenue NetIncome If Revenue = 1000 NetIncome will be about 2,307.70 Interpreting regression equation

51. Other Problems The expected cost for dinner for two couples (4 people) would be $95.06 Cost = 15.22 + 19.96 Persons If “Persons” = 4, what is the prediction for “Cost”? Cost = 15.22 + 19.96 Persons Cost = 15.22 + 19.96 (4) Cost = 15.22 + 79.84 = 95.06 Prediction line Y’ = a + b 1 X 1 Y-intercept Slope If “Persons” = 1, what is the prediction for “Cost”? Cost = 15.22 + 19.96 Persons Cost = 15.22 + 19.96 (1) Cost = 15.22 + 19.96 = 35.18 People Cost If People = 4 Cost will be about 95.06

52. Other Problems The expected cost for rent on an 800 square foot apartment is $990 Rent = 150 + 1.05 SqFt If “SqFt” = 800, what is the prediction for “Rent”? Rent = 150 + 1.05 SqFt Rent = 150 + 1.05 (800) Rent = 150 + 840 = 990 Prediction line Y’ = a + b 1 X 1 Y-intercept Slope Square Feet Cost If SqFt = 800 Rent will be about 990 If “SqFt” = 2500, what is the prediction for “Rent”? Rent = 150 + 1.05 SqFt Rent = 150 + 1.05 (2500) Rent = 150 + 840 = 2,775

53. Assumptions Underlying Linear Regression These Y values are normally distributed. The means of these normal distributions of Y values all lie on the straight line of regression. For each value of X, there is a group of Y values The standard deviations of these normal distributions are equal.

54. Is the regression line better than just guessing the mean of the Y variable? How much does the information about the relationship actually help? Which minimizes error better? How much better does the regression line predict the observed results? r2r2 Wow!

55. What is r 2 ? r 2 = The proportion of the total variance in one variable that is predictable by its relationship with the other variable If mother’s and daughter’s heights are correlated with an r =.8, then what amount (proportion or percentage) of variance of mother’s height is accounted for by daughter’s height? Examples.64 because (.8) 2 =.64

56. What is r 2 ? r 2 = The proportion of the total variance in one variable that is predictable for its relationship with the other variable If mother’s and daughter’s heights are correlated with an r =.8, then what proportion of variance of mother’s height is not accounted for by daughter’s height? Examples.36 because (1.0 -.64) =.36 or 36% because 100% - 64% = 36%

57. What is r 2 ? r 2 = The proportion of the total variance in one variable that is predictable for its relationship with the other variable If ice cream sales and temperature are correlated with an r =.5, then what amount (proportion or percentage) of variance of ice cream sales is accounted for by temperature? Examples.25 because (.5) 2 =.25

58. What is r 2 ? r 2 = The proportion of the total variance in one variable that is predictable for its relationship with the other variable If ice cream sales and temperature are correlated with an r =.5, then what amount (proportion or percentage) of variance of ice cream sales is not accounted for by temperature? Examples.75 because (1.0 -.25) =.75 or 75% because 100% - 25% = 75%

59. 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.