2. BNAD 276: Statistical Inference in Management Spring 2016 Green sheets

4. Before our fourth and final exam (April 28 th ) OpenStax Chapters 1 – 13 (Chapter 12 is emphasized) Plous Chapter 17: Social Influences Chapter 18: Group Judgments and Decisions Schedule of readings

5. On class website: Please complete homework worksheet #17 Hypothesis testing with Correlations Worksheet Due: Thursday, April 14 th Homework

6. By the end of lecture today 4/12/16 Logic of hypothesis testing with Correlations Interpreting the Correlations and scatterplots Simple Regression Using correlation for predictions

7. Exam 3 Thanks for your patience and cooperation We should have the grades up by Tuesday (takes about a week) It went really well!

8. Correlation Correlation: Measure of how two variables co-occur and also can be used for prediction Range between -1 and +1 Range between -1 and +1 The closer to zero the weaker the relationship and the worse the prediction The closer to zero the weaker the relationship and the worse the prediction Positive or negative Positive or negative Remember, We’ll call the correlations “r” Revisit this slide

9. Positive correlation Positive correlation: as values on one variable go up, so do values for other variable pairs of observations tend to occupy similar relative positions higher scores on one variable tend to co-occur with higher scores on the second variable lower scores on one variable tend to co-occur with lower scores on the second variable scatterplot shows clusters of point from lower left to upper right Remember, Correlation = “r” Revisit this slide

10. Negative correlation Negative correlation: as values on one variable go up, values for other variable go down pairs of observations tend to occupy dissimilar relative positions higher scores on one variable tend to co-occur with lower scores on the second variable lower scores on one variable tend to co-occur with higher scores on the second variable scatterplot shows clusters of point from upper left to lower right Remember, Correlation = “r” Revisit this slide

11. Zero correlation as values on one variable go up, values for the other variable go... anywhere pairs of observations tend to occupy seemingly random relative positions scatterplot shows no apparent slope Revisit this slide

12. Is it possible that they are causally related? Correlation does not imply causation Yes, but the correlational analysis does not answer that question What if it’s a perfect correlation – isn’t that causal? No, it feels more compelling, but is neutral about causality Number of Birthday Cakes Number of Birthdays Remember the birthday cakes! Revisit this slide

13. Correlation - How do numerical values change? r = +0.97 r = -0.48 r = -0.91 r = 0.61 Revisit this slide

14. Height of Daughters (inches) Height of Mothers (in) 48 52 56 60 64 68 72 76 48 52 5660 64 68 72 This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). Both axes and values are labeled Both axes have real numbers listed Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number) Revisit this slide

15. Height of Daughters (inches) Height of Mothers (in) 48 52 56 60 64 68 72 76 48 52 5660 64 68 72 This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). Both axes and values are labeled Both axes have real numbers listed Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number)

16. Height of Daughters (inches) Height of Mothers (in) 48 52 56 60 64 68 72 76 48 52 5660 64 68 72 This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). Both axes and values are labeled Both axes have real numbers listed Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number) Revisit this slide

17. Height of Daughters (inches) Height of Mothers (in) 48 52 56 60 64 68 72 76 48 52 5660 64 68 72 This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). Both axes and values are labeled Both axes have real numbers listed Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number) Revisit this slide

18. Height of Daughters (inches) Height of Mothers (in) 48 52 56 60 64 68 72 76 48 52 5660 64 68 72 This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). Both axes and values are labeled Both axes have real numbers listed Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number) Revisit this slide Statistically significant p < 0.05 Reject the null hypothesis

19. 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

20. 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

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

22. 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

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

24. Five steps to hypothesis testing Step 3: Calculations

25. Five steps to hypothesis testing Step 3: Calculations

26. 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

27. 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

28. 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

29. 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” Revisit this slide

30. 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

31. 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

32. 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?

33. 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?

34. 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?

35. What do we care about? Correlation matrices

36. 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

37. Correlation matrices What do we care about?

38. Correlation matrices What do we care about?

39. Correlation matrices What do we care about?

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

41. Correlation matrices What do we care about? Critical value from table r (148) = 0.195

42. 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?

43. 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

44. 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

45. 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. Revisit this slide

46. 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.

47. 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?

48. Predicting Restaurant Bill 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

49. Predicting Rent 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 + 2,625 = 2,775

50. 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.

51. Regression Example Do more sales calls result in more sales made? Dependent Variable Independent Variable Ethan Isabella Ava Emma Emily Jacob Joshua 60 70 0 1 2 3 4 Number of sales calls made Number of systems sold 10 20 30 40 50 0 Step 1: Draw scatterplot Step 2: Estimate r

52. Regression Example Do more sales calls result in more sales made? Step 3: Calculate r Step 4: Is it a significant correlation?

53. Do more sales calls result in more sales made? Step 4: Is it a significant correlation? n = 10, df = 8 alpha =.05 Observed r is larger than critical r (0.71 > 0.632) therefore we reject the null hypothesis. Yes it is a significant correlation r (8) = 0.71; p < 0.05 Step 3: Calculate r Step 4: Is it a significant correlation?

54. Regression: Predicting sales Step 1: Draw prediction line What are we predicting? r = 0.71 b = 11.579 (slope) a = 20.526 (intercept) Draw a regression line and regression equation

55. Regression: Predicting sales Step 1: Draw prediction line r = 0.71 b = 11.579 (slope) a = 20.526 (intercept) Draw a regression line and regression equation

56. Regression: Predicting sales Step 1: Draw prediction line r = 0.71 b = 11.579 (slope) a = 20.526 (intercept) Draw a regression line and regression equation

57. Step 2: State the regression equation Y’ = a + bx Y’ = 20.526 + 11.579x Step 3: Solve for some value of Y’ Y’ = 20.526 + 11.579(1) Y’ = 32.105 If make one sales call You should sell 32.105 systems Regression: Predicting sales Step 1: Predict sales for a certain number of sales calls What should you expect from a salesperson who makes 1 calls? Madison Joshua They should sell 32.105 systems If they sell more  over performing If they sell fewer  underperforming

58. Step 2: State the regression equation Y’ = a + bx Y’ = 20.526 + 11.579x Step 3: Solve for some value of Y’ Y’ = 20.526 + 11.579(2) Y’ = 43.684 Regression: Predicting sales Step 1: Predict sales for a certain number of sales calls What should you expect from a salesperson who makes 2 calls? If make two sales call You should sell 43.684 systems Isabella Jacob They should sell 43.68 systems If they sell more  over performing If they sell fewer  underperforming