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My last name starts with a letter somewhere between A. A – D B. E – L C. M – R D. S – Z

Please click in. Set your clicker to channel 41. My last name starts with a letter somewhere between A. A – D B. E – L C. M – R D. S – Z .

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My last name starts with a letter somewhere between A. A – D B. E – L C. M – R D. S – Z

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  1. Please click in Set your clicker to channel 41 My last name starts with a letter somewhere between A. A – D B. E – L C. M – R D. S – Z

  2. Introduction to Statistics for the Social SciencesSBS200, COMM200, GEOG200, PA200, POL200, SOC200Lecture Section 001, Fall, 2011Room 201 Physics-Atmospheric Sciences (PAS)10:00 - 10:50 Mondays & Wednesdays + Lab Session Welcome Please double check – All cell phones other electronic devices are turned off and stowed away http://www.youtube.com/watch?v=oSQJP40PcGI

  3. Use this as your study guide By the end of lecture today11/16/11 Logic of hypothesis testing with Correlations Interpreting the Correlations and scatterplots Simple Regression Using correlation for predictions r versus r2

  4. Readings for next exam 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

  5. Extra Credit Opportunity Design a question/topic Gather Data Present data in a memo

  6. Homework due next class November 21st Assignment 13: Regression worksheet (can be found on class website) Homework Questions? Please double check – All cell phones other electronic devices are turned off and stowed away

  7. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses For correlation null is that r = 0 (no relationship) Step 2: Decision rule • Alpha level? (α= .05 or .01)? • Critical statistic (e.g. critical r) value from table? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed r is bigger then critical r then reject null Step 5: Conclusion - tie findings back in to research problem

  8. Finding a statistically significant correlation • The result is “statistically significant” if: • the observed correlation is larger than the critical correlationwe 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

  9. 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 What are we predicting? What are we predicting? Dependent Variable Dependent Variable Independent Variable Independent Variable

  10. What are we predicting? Correlation Positive correlation: as values on one variable go up, so do values for the other variable Negative correlation: as values on one variable go up, the values for the other variable go down Yearly income by expenses per year YearlyIncome Positive Correlation Expenses per year

  11. What are we predicting? Correlation Positive correlation: as values on one variable go up, so do values for the other variable Negative correlation: as values on one variable go up, the values for the other variable go down Temperatures by time spent outside in Tucson in summer Temperature Negative Correlation Timeoutside

  12. What are we predicting? Correlation Positive correlation: as values on one variable go up, so do values for the other variable Negative correlation: as values on one variable go up, the values for the other variable go down Height by average driving speed Height Zero Correlation Average Speed

  13. What are we predicting? Correlation Positive correlation: as values on one variable go up, so do values for the other variable Negative correlation: as values on one variable go up, the values for the other variable go down Amount Healthtex spends per month on advertising by sales in the month Amountof sales Positive Correlation Amount spent On Advertising

  14. YearlyIncome Expenses per year Correlation - What do we need to define a line If you probably make this much Y-intercept = “a” (also “b0”)Where the line crosses the Y axis Slope = “b” (also “b1”)How steep the line is If you spend 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

  15. Yearly Income Yearly Income YearlyIncome Expenses per year Expenses per year Expenses per year Correlation - What do we need to define a line Y-intercept = “a”Where the line crosses the Y axis Slope = “b” How steep the line is Y-intercept is good…slope is wrong Y-intercept is wrong…slope is good

  16. BrushingTeeth BrushingTeeth BrushingTeeth NumberCavities NumberCavities NumberCavities Correlation - What do we need to define a line Y-intercept = “a”Where the line crosses the Y axis Slope = “b” How steep the line is Y-intercept is wrong…slope is good Y-intercept is good…slope is wrong

  17. Assumptions Underlying Linear Regression • For each value of X, there is a group of Y values • These Y values are normally distributed. • The means of these normal distributions of Y values all lie on the straight line of regression. • The standard deviations of these normal distributions are equal.

  18. Correlation - the prediction line - what is it good for? 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

  19. 5 4 Number of times per day teeth are brushed 3 2 1 0 0 1 2 3 4 5 Number of cavities Correlation - let’s do another one Does brushing your teeth correlate with fewer cavities? Step 1: Draw scatterplot Step 2: Data table X Y XY X2 Y2 1 5 5 1 25 3 4 12 9 16 2 3 6 4 9 3 2 6 9 4 5 1 5 25 1 Σ 14 15 34 48 55 Step 3: Estimate r and prediction line Step 4: Find r

  20. Correlation - Let’s do one Step 1: Find n n = 5 (5 pairs) Step 2: Find ΣX and ΣY Step 3: Find ΣXY Step 4: Find ΣX2 and ΣY2 Step 5: Plug in the numbers X Y XY X2 Y2 1 5 5 1 25 3 4 12 9 16 2 3 6 4 9 3 2 6 9 4 5 1 5 25 1 Σ 14 15 34 48 55 The formula:

  21. r = r = [√[(5)(55)-(15)2] [√[(5)(48)-(14)2] - 40 (170 - 210) = [√50 ] [√44 ] 46.90 Correlation - Let’s do one Step 1: Find n n = 5 (5 pairs) Step 2: Find ΣX and ΣY Step 3: Find ΣXY Step 4: Find ΣX2 and ΣY2 Step 5: Plug in the numbers (5)(34)-(14)(15) X Y XY X2 Y2 1 5 5 1 25 3 4 12 9 16 2 3 6 4 9 3 2 6 9 4 5 1 5 25 1 Σ 14 15 34 48 55 r = -.85 The formula:

  22. X Y XY X2 Y2 . 1 5 5 1 25 3 4 12 9 16 2 3 6 4 9 3 2 6 9 4 5 1 5 25 1 Σ 14 15 34 48 55 Find r r = -0.85

  23. X Y XY X2 Y2 . 1 5 5 1 25 3 4 12 9 16 2 3 6 4 9 3 2 6 9 4 5 1 5 25 1 Σ 14 15 34 48 55 Draw a scatterplot

  24. X Y XY X2 Y2 1 5 5 1 25 3 4 12 9 16 2 3 6 4 9 3 2 6 9 4 5 1 5 25 1 Σ 14 15 34 48 55 Draw a scatterplot

  25. Draw a regression line

  26. Draw a regression line

  27. r = -0.85 b= - 0.91(slope) a= 5.5 (intercept) Draw a regression line and regression equation

  28. Prediction line Y’ = a+ b1X1 Y’ = 842 + (-37.5)X1 Interpreting regression equation Y-intercept a) Interpret the slope of the fitted regression line:Sales = 842 – 37.5 Price Slope Notice in this case it is negative A slope of “37.5” suggests that raising “price” by 1 unit will reduce “sales” by 37.5 units b) If “price” = 20, what is the prediction for “Sales”?Sales = 842 – 37.5 Price Sales = 842 - 37.5 Price Sales = 842 - (37.5) (20) Sales = 842 - (37.5) (20) = 842 – 750 = 92 Sales price of product

  29. Prediction line Y’ = a+ b1X1 Y’ = 842 + (-37.5)X1 Interpreting regression equation Y-intercept a) Interpret the slope of the fitted regression line:Sales = 842 – 37.5 Price Slope A slope of “37.5” suggests that raising “price” by 1 unit will reduce “sales” by 37.5 units b) If “price” = 20, what is the prediction for “Sales”?Sales = 842 – 37.5 Price Sales = 842 - 37.5 Price Sales = 842 - (37.5) (20) Sales = 842 - (37.5) (20) = 842 – 750 = 92 (20, 92) Sales probablyabout 92 units Sales price of product If Price = 20

  30. Prediction line Y’ = a+ b1X1 Y’ = 2.277 + (.0307)X1 Interpreting regression equation a) The regression equation: NetIncome = 2,277 + .0307 Revenue Interpret the slope Y-intercept Slope Notice in this case it is positive A slope of “.0307” suggests that raising “Revenue” by 1 dollar, NetIncome will raise by 3 cents b) If “Revenue” = 1,000, what is the prediction for “NetIncome”? NetIncome = 2,277 + .0307 Revenue NetIncome = 2,277 + (.0307 )(1,000) NetIncome = 2,277 + 30.7 = 2,307.7 (1,000, 2,307.7) NetIncome Revenue

  31. Prediction line Y’ = a+ b1X1 Y’ = 2,277 + (.0307)X1 Interpreting regression equation a) The regression equation: NetIncome = 2,277 + .0307 Revenue Interpret the slope Y-intercept Slope A slope of “.0307” suggests that raising “Revenue” by 1 dollar, NetIncome will raise by 3 cents b) If “Revenue” = 1,000, what is the prediction for “NetIncome”? NetIncome will be about 2,307.70 NetIncome = 2,277 + .0307 Revenue NetIncome = 2,277 + (.0307 )(1,000) NetIncome = 2,277 + 30.7 = 2,307.7 (1,000, 2,307.7) NetIncome Revenue If Revenue = 1000

  32. Prediction line Y’ = a+ b1X1 Other Problems Cost will be about 95.06 Cost Y-intercept The expected cost for dinner for two couples (4 people) would be $95.06Cost = 15.22 + 19.96 Persons People Slope If People = 4 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 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

  33. Prediction line Y’ = a+ b1X1 Other Problems Rent will be about 990 Cost Y-intercept Slope Square Feet If SqFt = 800 The expected cost for rent on an 800 square foot apartment is $990Rent = 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 If “SqFt” = 2500, what is the prediction for “Rent”? Rent = 150 + 1.05 SqFt Rent = 150 + 1.05 (2500) Rent = 150 + 840 = 2,775

  34. Prediction line Y’ = a+ b1X1 Frequency of Teeth brushing will be about Other Problems Y-intercept If number of cavities = 3 Slope The expected frequeny of teeth brushing for having one cavity is Frequency of teeth brushing= 5.5 + (-.91) Cavities If “Cavities” = 3, what is the prediction for “Frequency of teeth brushing”? Frequency of teeth brushing= 5.5 + (-.91) Cavities Frequency of teeth brushing= 5.5 + (-.91) (3) Frequency of teeth brushing= 5.5 + (-2.73) = 2.77 (3.0, 2.77)

  35. Draw a regression line and regression equation Prediction line Y’ = b1X1+ b0 Y’ = (-.91)X 1+ 5.5 b0 = 5.5 (intercept) b1 = - 0.91(slope) r = - 0.85

  36. 5 4 Number of times per day teeth are brushed 3 2 1 0 0 1 2 3 4 5 Number of cavities Prediction line Y’ = b1X 1+ b0 Y’ = (-.91)X 1+ 5.5 Correlation - Evaluating the prediction line Does the prediction line perfectly predict the Ys from the Xs? No, let’s see How much “error” is there? Exactly? Residuals The green lines show how much “error” there is in our prediction line…how much we are wrong in our predictions

  37. 5 4 3 Number of times per day teeth are brushed 2 1 0 0 1 2 3 4 5 Number of cavities A note about curvilinear relationships and patterns of the residuals How well does the prediction line predict the Ys from the Xs? Residuals • 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)

  38. 5 4 Number of times per day teeth are brushed 3 2 1 0 0 1 2 3 4 5 Number of cavities How well does the prediction line predict the Ys from the Xs? Residuals • Slope doesn’t give “variability” info • Intercept doesn’t give “variability info • Correlation “r” does give “variability info • Residuals do give “variability info

  39. Sound familiar?? What if we want to know the “average deviation score”? Finding the standard error of the estimate (line) 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

  40. Which minimizes errorbetter? 5 4 Number of times per day teeth are brushed 3 2 1 0 0 1 2 3 4 5 Number of cavities Is the regression line better than just guessing the mean of the Y variable?How much does the information about the relationship actually help? 5 4 # of times teeth are brushed 3 2 1 0 0 1 2 3 4 5 Number of cavities How much better does the regression line predict the observed results? r2 Wow!

  41. What is r2? r2 = The proportion of the total variance in one variable that is predictable by its relationship with the other variable

  42. What is r2? r2 = The proportion of the total variance in one variable that is predictable by its relationship with the other variable Examples 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? .64 because (.8)2 = .64 or 64% because (8%)2 = 64%

  43. What is r2? r2 = The proportion of the total variance in one variable that is predictable for its relationship with the other variable Examples If mother’s and daughter’s heights are correlated with an r = .8, then what proportion of variance of mother’s height is not accountedfor by daughter’s height? .36 because (1.0 - .64) = .36 or 36% because 100% - 64% = 36%

  44. What is r2? r2 = The proportion of the total variance in one variable that is predictable for its relationship with the other variable Examples 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? .25 because (.5)2 = .25 or 25% because (5%)2 = 25%

  45. What is r2? r2 = The proportion of the total variance in one variable that is predictable for its relationship with the other variable Examples 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 accountedfor by temperature? .75 because (1.0 - .25) = .75 or 75% because 100% - 25% = 75%

  46. Thank you! See you next time!!

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