2. Modern Languages 14131211109 87 6 54321 111098765 43 2 Row A Row B Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 212019181716 1514 13 12111098 212019181716 13 12111098 141312 table 7 6 54321 Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 321 21 1413 Projection Booth 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 7 6 5432 1 765 43 2 1 7 6 5432 1 765 43 2 1 7 6 54321 765 43 2 1 7 6 54321 765 43 2 1 7 6 54321 table Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 321 28 27 26252423 22 282726 2524 23 22 282726 2524 23 22 28 27 26252423 22 282726 2524 23 22 282726 2524 23 22 28 27 26252423 22 282726 2524 23 22 282726 2524 23 22 282726 2524 23 22 R/L handed broken desk Stage Lecturer’s desk Screen 1

3. MGMT 276: Statistical Inference in Management Spring 2015

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

5. Remember… In a negatively skewed distribution: mean < median < mode Score on Exam Note: Label and Numbers Mean Frequency Mode 87.5 = mode = tallest point 85 = median = middle score 80 = mean = balance point Median Note: Always “frequency”

7. Before our fourth exam (April 30 th ) Lind Chapter 13: Linear Regression and Correlation Chapter 14: Multiple Regression Chapter 15: Chi-Square Plous Chapter 17: Social Influences Chapter 18: Group Judgments and Decisions Schedule of readings

8. Logic of hypothesis testing with Correlations Interpreting the Correlations and scatterplots Simple and Multiple Regression Using correlation for predictions r versus r2 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 “r2” (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 4/16/15

9. On class website: Please print and complete homework worksheet #16 Interpreting Correlation and Simple Regression Homework due – Tuesday (April 21 st )

10. 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

11. 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

12. 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

13. 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

14. 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

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

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)

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

19. 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

20. 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

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. 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

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

24. 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

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

26. Five steps to hypothesis testing Step 3: Calculations

27. Five steps to hypothesis testing Step 3: Calculations

28. 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

29. 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

30. 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

31. 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”

32. 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

33. 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

34. 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

35. 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?

36. 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?

37. 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?

38. What do we care about? Correlation matrices

39. 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

40. Correlation matrices What do we care about?

41. Correlation matrices What do we care about?

42. Correlation matrices What do we care about?

43. 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?

44. 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

45. 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

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. 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?

49. 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

50. 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 + 840 = 2,775