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 12/2/15

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 23: Multiple Regression Using Excel Due: Friday, December 4 th

9. 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.

10. 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

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

12. 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?

13. 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

14. 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

15. 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

16. Rory’s Regression: Predicting sales from number of visits (sales calls) Regression line (and equation) r = 0.71 b = 11.579 (slope) a = 20.526 (intercept) Predict using regression line (and regression equation) Slope: as sales calls increase by 1, sales should increase by 11.579 Describe relationship Correlation: This is a strong positive correlation. Sales tend to increase as sales calls increase Intercept: suggests that we can assume each salesperson will sell at least 20.526 systems Review Dependent Variable Independent Variable

17. 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

18. 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

19. 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(3) Y’ = 55.263 Regression: Predicting sales Step 1: Predict sales for a certain number of sales calls What should you expect from a salesperson who makes 3 calls? If make three sales call You should sell 55.263 systems Ava Emma They should sell 55.263 systems If they sell more  over performing If they sell fewer  underperforming

20. Step 2: State the regression equation Y’ = a + bx Y’ = 20.526 + 11.579x Regression: Predicting sales Step 1: Predict sales for a certain number of sales calls What should you expect from a salesperson who makes 4 calls? Step 3: Solve for some value of Y’ Y’ = 20.526 + 11.579(4) Y’ = 66.842 If make four sales calls You should sell 66.84 systems Emily They should sell 66.84 systems If they sell more  over performing If they sell fewer  underperforming

21. Regression: Evaluating Staff Step 1: Compare expected sales levels to actual sales levels What should you expect from each salesperson They should sell x systems depending on sales calls If they sell more  over performing If they sell fewer  underperforming Madison Isabella Ava Emma Emily Jacob Joshua

22. Regression: Evaluating Staff Step 1: Compare expected sales levels to actual sales levels How did Ava do? Ava sold 14.7 more than expected taking into account how many sales calls she made  over performing Ava 14.7 Difference between expected Y’ and actual Y is called “residual” (it’s a deviation score) 70-55.3=14.7

23. Regression: Evaluating Staff Step 1: Compare expected sales levels to actual sales levels How did Jacob do? Jacob sold 23.684 fewer than expected taking into account how many sales calls he made  under performing Ava -23.7 Difference between expected Y’ and actual Y is called “residual” (it’s a deviation score) Jacob 20-43.7=-23.7

24. Regression: Evaluating Staff Step 1: Compare expected sales levels to actual sales levels What should you expect from each salesperson They should sell x systems depending on sales calls If they sell more  over performing If they sell fewer  underperforming Madison Isabella Ava Emma Emily Jacob Joshua

25. Regression: Evaluating Staff Step 1: Compare expected sales levels to actual sales levels Madison Isabella Ava Emma Emily Jacob Joshua 14.7 Difference between expected Y’ and actual Y is called “residual” (it’s a deviation score) -23.7 -6.8 7.9

26. 14.7 Difference between expected Y’ and actual Y is called “residual” (it’s a deviation score) Does the prediction line perfectly the predicted variable when using the predictor variable? The green lines show how much “error” there is in our prediction line…how much we are wrong in our predictions How would we find our “average residual”? No, we are wrong sometimes… How can we estimate how much “error” we have? Exactly? -23.7

27. Difference between expected Y’ and actual Y is called “residual” (it’s a deviation score) How do we find the average amount of error in our prediction The green lines show how much “error” there is in our prediction line…how much we are wrong in our predictions How would we find our “average residual”? Step 1: Find error for each value (just the residuals) Y – Y’ Ava is 14.7 Emily is -6.8 Madison is 7.9 Jacob is -23.7 Residual scores The average amount by which actual scores deviate on either side of the predicted score N ΣxΣx Big problem Σ (Y – Y’) = 0 2 Square the deviations Step 2: Add up the residuals Σ (Y – Y’) Divide by df 2 n - 2 Σ (Y – Y’) Square root

28. Difference between expected Y’ and actual Y is called “residual” (it’s a deviation score) How do we find the average amount of error in our prediction The green lines show how much “error” there is in our prediction line…how much we are wrong in our predictions How would we find our “average residual”? Step 1: Find error for each value (just the residuals) Y – Y’ Step 2: Find average ∑(Y – Y’) 2 n - 2 √ Diallo is 0” Mike is -4” Hunter is -2 Preston is 2” Deviation scores N ΣxΣx Sound familiar??

29. These would be helpful to know by heart – please memorize these formula Standard error of the estimate (line) =

30. Slope doesn’t give “variability” info Intercept doesn’t give “variability info Correlation “r” does give “variability info How well does the prediction line predict the predicted variable when using the predictor variable? Residuals do give “variability info Standard error of the estimate (line) What if we want to know the “average deviation score”? Finding the standard error of the estimate (line) Standard error of the estimate: a measure of the average amount of predictive error the average amount that Y’ scores differ from Y scores a mean of the lengths of the green lines

31. Shorter green lines suggest better prediction – smaller error Longer green lines suggest worse prediction – larger error Why are green lines vertical? Remember, we are predicting the variable on the Y axis So, error would be how we are wrong about Y (vertical) How well does the prediction line predict the Ys from the Xs? Residuals A note about curvilinear relationships and patterns of the residuals

32. 14.7 Difference between expected Y’ and actual Y is called “residual” (it’s a deviation score) Does the prediction line perfectly the predicted variable when using the predictor variable? The green lines show how much “error” there is in our prediction line…how much we are wrong in our predictions No, we are wrong sometimes… How can we estimate how much “error” we have? -23.7 Perfect correlation = +1.00 or -1.00 Each variable perfectly predicts the other No variability in the scatterplot The dots approximate a straight line

33. Regression Analysis – Least Squares Principle When we calculate the regression line we try to: minimize distance between predicted Ys and actual (data) Y points (length of green lines) remember because of the negative and positive values cancelling each other out we have to square those distance (deviations) so we are trying to minimize the “sum of squares of the vertical distances between the actual Y values and the predicted Y values”

34. 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!

35. 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

36. 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%

37. 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

38. 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%

39. Some useful terms 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)

40. Rory’s Regression: Predicting sales from number of visits (sales calls) Regression line (and equation) r = 0.71 b = 11.579 (slope) a = 20.526 (intercept) Predict using regression line (and regression equation) Slope: as sales calls increase by 1, sales should increase by 11.579 Describe relationship Correlation: This is a strong positive correlation. Sales tend to increase as sales calls increase Intercept: suggests that we can assume each salesperson will sell at least 20.526 systems Review Dependent Variable Independent Variable

41. Review

42. Summary Slope: as sales calls increase by one, 11.579 more systems should be sold Intercept: suggests that we can assume each salesperson will sell at least 20.526 systems Review