2. 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

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

5. By the end of lecture today 4/11/16 Project 4 Logic of hypothesis testing with Correlations Interpreting the Correlations and scatterplots Simple Regression Using correlation for predictions

6. On class website: Please complete homework worksheet #23 Correlations Worksheet Due: Wednesday, April 13 th Homework

7. 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

8. Project 4 - Two Correlations - We will use these to create two regression analyses

12. Everyone will want to be enrolled in one of the lab sessions Labs will meet this week Project 4

13. 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

14. 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

15. 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

16. 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

17. 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

18. Correlation - How do numerical values change? r = +0.97 r = -0.48 r = -0.91 r = 0.61 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)

21. 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

22. 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

23. 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

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. 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

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

27. 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

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

29. Five steps to hypothesis testing Step 3: Calculations

30. Five steps to hypothesis testing Step 3: Calculations

31. 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

32. 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

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. 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

35. 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

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 X Correlation of Y with Y Correlation of Z with Z Correlation matrices

37. 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?

38. 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?

39. 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?

40. What do we care about? Correlation matrices

41. 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

42. Correlation matrices What do we care about?

43. Correlation matrices What do we care about?

44. Correlation matrices What do we care about?

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

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