2. Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Fall 2015 Room 150 Harvill Building 10:00 - 10:50 Mondays, Wednesdays & Fridays. http://courses.eller.arizona.edu/mgmt/delaney/d15s_database_weekone_screenshot.xlsx

4. No Labs this week

5. 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) Over next couple of lectures 11/30/15

6. Before our next exam (December 7 th ) OpenStax Chapters 1 – 13 (Chapter 12 is emphasized) Plous Chapter 17: Social Influences Chapter 18: Group Judgments and Decisions Schedule of readings

8. Homework Assignment Go to D2L - Click on “Interactive Online Homework Assignments” Complete Assignment 22: Simple Regression Using Excel Due: Wednesday, December 2 nd

9. 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

10. 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

11. 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

12. 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

13. 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 Revisit this slide

14. 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

15. 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

16. 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

17. 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?

18. 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?

19. 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?

20. What do we care about? Correlation matrices

21. 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

22. Correlation matrices What do we care about?

23. Correlation matrices What do we care about?

24. Correlation matrices What do we care about?

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

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

27. 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?

28. 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

29. 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

30. 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

31. 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.

32. 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?

33. 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

34. 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