1. Unit 3 Correlation

2. Homework Assignment For the A: 1, 5, 7,11, 13, 14 - 18, 21, 27 - 32, 35, 37, 39, 41, 43, 45, 47 – 51, 55, 58, 59, 61, 63, 65, 69, 71 - 78 R1 – R6 For the C: 1, 7, 13, 21, 32, 35, 39, 43, 47 – 51, 55, 58, 61, 65, 69, 71 – 78 R1 – R6 For the D- : 1, 7, 21, 32, 39, 43, 47 – 51, 55, 58, 65, 69, R1 – R6 All problems must be complete, including explanations with complete sentences and or work to show if the question asks for it.

3. The Guessing Game Activity

4. Suppose we found the age and weight of a sample of 10 adults. Create a scatterplot of the data below. Is there any relationship between the age and weight of these adults? Age 24304128504649352039 Wt256124320185158129103196110130

5. Suppose we found the height and weight of a sample of 10 adults. Create a scatterplot of the data below. Is there any relationship between the height and weight of these adults? Ht74657772686062736164 Wt256124320185158129103196110130 Is it positive or negative? Weak or strong?

6. The closer the points in a scatterplot are to a straight line - the stronger the relationship. The farther away from a straight line – the weaker the relationship

7. positive negativeno Identify as having a positive association, a negative association, or no association. 1.Heights of mothers & heights of their adult daughters + 2. Age of a car in years and its current value 3.Weight of a person and calories consumed 4.Height of a person and the person’s birth month 5.Number of hours spent in safety training and the number of accidents that occur - + NO -

8. Self Check #14

9. Correlation Coefficient (r)- quantitativeA quantitative assessment of the strength & direction of the linear relationship between bivariate, quantitative data Pearson’s sample correlation is used most parameter -  rho) statistic - r

10. Calculate r. Interpret r in context. Speed Limit (mph) 555045403020 Avg. # of accidents (weekly) 28252117116 There is a strong, positive, linear relationship between speed limit and average number of accidents per week.

11. Moderate Correlation Strong correlation Properties of r (correlation coefficient) legitimate values of r is [-1,1] No Correlation Weak correlation

12. unitvalue of r does not depend on the unit of measurement for either variable x (in mm) 12152132261924 y 4710149812 Find r. Change to cm & find r. The correlations are the same.

13. value of r does not depend on which of the two variables is labeled x x 12152132261924 y4710149812 Switch x & y & find r. The correlations are the same.

14. non-resistantvalue of r is non-resistant x 12152132261924 y4710149822 Find r. Outliers affect the correlation coefficient

15. Internet Regression Activity

16. linearlyvalue of r is a measure of the extent to which x & y are linearly related A value of r close to zero does not rule out any strong relationship between x and y. definite r = 0, but has a definite relationship!

17. Methodist Ministers vs. Cuban Rum Give handout

18. Minister data: r =.9999 cause So does an increase in ministers cause an increase in consumption of rum?

19. Correlation does not imply causation

20. Self Check #15

21. Multiple Choice Test #5

22. Assignment #6

23. Least Square Regression Line LSRL

24. Bivariate data x – variable: is the independent or explanatory variable y- variable: is the dependent or response variable Use x to predict y

25. Least Squares Regression Line LSRL bestThe line that gives the best fit to the data set minimizesThe line that minimizes the sum of the squares of the deviations from the line

26. Sum of the squares = 61.25 -4 4.5 -5 y =.5(0) + 4 = 4 0 – 4 = -4 (0,0) (3,10) (6,2) (0,0) y =.5(3) + 4 = 5.5 10 – 5.5 = 4.5 y =.5(6) + 4 = 7 2 – 7 = -5

27. (0,0) (3,10) (6,2) Sum of the squares = 54 Use a calculator to find the line of best fit Find y - y -3 6 What is the sum of the deviations from the line? Will it always be zero? minimizes LSRL The line that minimizes the sum of the squares of the deviations from the line is the LSRL.

28. Regression Activity

29. Slope: unitx increase/decrease by For each unit increase in x, there is an approximate average increase/decrease of b in y. Interpretations Correlation coefficient: direction, strength, linear xy There is a direction, strength, linear of association between x and y.

30. The ages (in months) and heights (in inches) of seven children are given. x1624426075102120 y24303540485660 Find the LSRL. Interpret the slope and correlation coefficient in the context of the problem.

31. Correlation coefficient: strong, positive, linear age and height of children There is a strong, positive, linear association between the age and height of children. Slope: age of one month increase.34 inches in heights of children. For an increase in age of one month, there is an approximate increase of.34 inches in heights of children.

32. The ages (in months) and heights (in inches) of seven children are given. x1624426075102120 y24303540485660 Predict the height of a child who is 4.5 years old. Predict the height of someone who is 20 years old.

33. Extrapolation should notThe LSRL should not be used to predict y for values of x outside the data set. It is unknown whether the pattern observed in the scatterplot continues outside this range.

34. The ages (in months) and heights (in inches) of seven children are given. x1624426075102120 y24303540485660 Calculate x & y. Plot the point (x, y) on the LSRL. Will this point always be on the LSRL?

35. Disk Game

36. Regression Assignment

37. non-resistant The correlation coefficient and the LSRL are both non-resistant measures.

38. Formulas – on chart

39. The following statistics are found for the variables posted speed limit and the average number of accidents. Find the LSRL & predict the number of accidents for a posted speed limit of 50 mph.

40. Matching Descriptions to Scatter Plots

41. Self Check #16

42. Regression Assignment #2

43. Residuals, Residual Plots, & Influential points

44. Residual plot A scatterplot of the (x, residual) pairs. Residuals can be graphed against other statistics besides x linear associationPurpose is to tell if a linear association exist between the x & y variables no pattern linearIf no pattern exists between the points in the residual plot, then the association is linear.

45. Linear Not linear

46. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 One measure of the success of knee surgery is post-surgical range of motion for the knee joint following a knee dislocation. Is there a linear relationship between age & range of motion? Sketch a residual plot. Since there is no pattern in the residual plot, there is a linear relationship between age and range of motion x Residuals

47. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 Plot the residuals against the y- hats. How does this residual plot compare to the previous one? Residuals

48. Residual plots are the same no matter if plotted against x or y-hat. x Residuals

49. Coefficient of determination- r 2 variationygives the proportion of variation in y that can be attributed to an approximate linear relationship between x & y remains the same no matter which variable is labeled x

50. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 Let’s examine r 2. Suppose you were going to predict a future y but you didn’t know the x-value. Your best guess would be the overall mean of the existing y’s. SS y = 1564.917 Sum of the squared residuals (errors) using the mean of y.

51. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 Now suppose you were going to predict a future y but you DO know the x-value. Your best guess would be the point on the LSRL for that x-value (y-hat). Sum of the squared residuals (errors) using the LSRL. SS y = 1085.735

52. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 By what percent did the sum of the squared error go down when you went from just an “overall mean” model to the “regression on x” model? SS y = 1085.735 SS y = 1564.917 This is r 2 – the amount of the variation in the y-values that is explained by the x-values.

53. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 How well does age predict the range of motion after knee surgery? Approximately 30.6% of the variation in range of motion after knee surgery can be explained by the linear regression of age and range of motion.

54. Interpretation of r 2 r 2 % y xy Approximately r 2 % of the variation in y can be explained by the LSRL of x & y.

55. Computer-generated regression analysis of knee surgery data: PredictorCoefStdevTP Constant107.5811.129.670.000 Age0.87100.41462.100.062 s = 10.42R-sq = 30.6%R-sq(adj) = 23.7% What is the equation of the LSRL? Find the slope & y-intercept. NEVER use adjusted r 2 ! before Be sure to convert r 2 to decimal before taking the square root! What are the correlation coefficient and the coefficient of determination?

56. Residual Notes and Assignment

57. Outlier – largeIn a regression setting, an outlier is a data point with a large residual

58. Influential point- A point that influences where the LSRL is located If removed, it will significantly change the slope of the LSRL

59. RacketResonance Acceleration (Hz) (m/sec/sec) 110536.0 210635.0 311034.5 411136.8 511237.0 611334.0 711334.2 811433.8 911435.0 1011935.0 1112033.6 1212134.2 1312636.2 1418930.0 One factor in the development of tennis elbow is the impact-induced vibration of the racket and arm at ball contact. Sketch a scatterplot of these data. Calculate the LSRL & correlation coefficient. Does there appear to be an influential point? If so, remove it and then calculate the new LSRL & correlation coefficient.

60. (189,30) could be influential. Remove & recalculate LSRL

61. (189,30) was influential since it moved the LSRL

62. Which of these measures are resistant? LSRL Correlation coefficient Coefficient of determination NONE NONE – all are affected by outliers

63. Self Check #17

64. Multiple Choice Test #6

65. Assignment #7