1. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Ten Regression and Correlation

2. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Scatter Diagram a plot of paired data to determine or show a relationship between two variables

3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 Paired Data

4. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 Scatter Diagram

5. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 Linear Correlation The general trend of the points seems to follow a straight line segment.

6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 Linear Correlation

7. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 Non-Linear Correlation

8. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 No Linear Correlation

9. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 High Linear Correlation Points lie close to a straight line.

10. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 High Linear Correlation

11. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 Moderate Linear Correlation

12. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Low Linear Correlation

13. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Perfect Linear Correlation

14. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Questions Arising Can we find a relationship between x and y? How strong is the relationship?

15. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 When there appears to be a linear relationship between x and y: attempt to “fit” a line to the scatter diagram.

16. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 When using x values to predict y values: Call x the explanatory variable Call y the response variable

17. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 The Least Squares Line The sum of the squares of the vertical distances from the points to the line is made as small as possible.

18. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Least Squares Criterion The sum of the squares of the vertical distances from the points to the line is made as small as possible.

19. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 Equation of the Least Squares Line y = a + bx a = the y-interceptb = the slope

20. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 The equation of the least squares line is: y = a + bx y = 2.8 + 1.7x

21. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 The following point will always be on the least squares line:

22. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 Graphing the least squares line Using two values in the range of x, compute two corresponding y values. Plot these points. Join the points with a straight line.

23. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 Graphing y = 30.9 + 1.7x Use (8.3, 16.9) (average of the x’s, the average of the y’s) Try x = 5. Compute y: y = 2.8 + 1.7(5)= 11.3

24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 Sketching the Line Using the Points (8.3, 16.9) and (5, 11.3)

25. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 Using the Equation of the Least Squares Line to Make Predictions Choose a value for x (within the range of x values). Substitute the selected x in the least squares equation. Determine corresponding value of y.

26. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 Predict the time to make a trip of 14 miles Equation of least squares line: y = 2.8 + 1.7x Substitute x = 14: y = 2.8 + 1.7 (14) y = 26.6 According to the least squares equation, a trip of 14 miles would take 26.6 minutes.

27. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 Interpolation Using the least squares line to predict y values for x values that fall between the points in the scatter diagram

28. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 Extrapolation Prediction beyond the range of observations

29. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 The least squares line and prediction, y p : y = a + bx y = 2.8 + 1.7x For x = 8, y p = 2.8 + 1.7(8) = 16.4

30. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 Try not to use the least squares line to predict y values for x values beyond the data extremes of the sample x distribution.

31. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 The Linear Correlation Coefficient, r A measurement of the strength of the linear association between two variables Also called the Pearson product-moment correlation coefficient

32. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 y x Positive Correlation

33. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 y x Negative Correlation

34. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 y x Little or No Linear Correlation

35. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 What type of correlation is expected? Height and weight Mileage on tires and remaining tread IQ and height Years of driving experience and insurance rates

36. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 Linear correlation coefficient  1  r  +1

37. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 37 y x If r = 0, scatter diagram might look like:

38. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 38 y x If r = +1, all points lie on the least squares line

39. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 39 y x If r = –1, all points lie on the least squares line

40. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 40 y x – 1 < r < 0

41. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 41 y x 0 < r < 1

42. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 42 The Correlation Coefficient, r = 0.9753643 r  0.98

43. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 43 A statistic related to r: the coefficient of determination = r 2

44. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 44 Coefficient of Determination a measure of the proportion of the variation in y that is explained by the regression line using x as the predicting variable

45. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 45 Interpretation of r 2 If r = 0.9753643, then what percent of the variation in minutes (y) is explained by the linear relationship with x, miles traveled? What percent is explained by other causes?

46. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 46 Interpretation of r 2 If r = 0.9753643, then r 2 =.9513355 Approximately 95 percent of the variation in minutes (y) is explained by the linear relationship with x, miles traveled. Less than five percent is explained by other causes.

47. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 47 Warning The correlation coefficient ( r) measures the strength of the relationship between two variables. Just because two variables are related does not imply that there is a cause-and- effect relationship between them.

48. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 48 Testing the Correlation Coefficient Determining whether a value of the sample correlation coefficient, r, is far enough from zero to indicate correlation in the population.

49. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 49 The Population Correlation Coefficient  = Greek letter “rho”

50. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 50 Hypotheses to Test Rho Assume that both variables x and y are normally distributed. To test if the (x, y) values are correlated in the population, set up the null hypothesis that they are not correlated: H 0 :x and y are not correlated, so  = 0.

51. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 51 H 0 :  = 0 If you believe  is positive, use a right-tailed test. H 1 :  > 0

52. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 52 H 0 :  = 0 If you believe  is negative, use a left-tailed test. H 1 :  < 0

53. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 53 H 0 :  = 0 If you believe  is not equal to zero, use a two-tailed test. H 1 :   0

54. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 54 Convert r to a Student’s t Distribution

55. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 55 A researcher wishes to determine (at 5% level of significance) if there is a positive correlation between x, the number of hours per week a child watches television and y, the cholesterol measurement for the child. Assume that both x and y are normally distributed.

56. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 56 Correlation Between Hours of Television and Cholesterol Suppose that a sample of x and y values for 25 children showed the correlation coefficient, r to be 0.42. Use a right-tailed test. The null hypothesis: H 0 :  = 0 The alternate hypothesis: H 1 :  > 0  = 0.05

57. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 57 Convert the sample statistic r = 0.42 to t using n = 25

58. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 58 Find critical t value for right- tailed test with  = 0.05 Use Table 6. d.f. = 25 - 2 = 23. t = 1.714 2.22 > 1.714 Reject the null hypothesis. Conclude that there is a positive correlation between the variables.

59. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 59 P Value Approach Use Table 6 in Appendix II, d.f. = 23 Our t value =2.22 is between 2.069 and 2.500. This gives P between 0.025 and 0.010. Since we would reject H 0 for any   P, we reject H 0 for  = 0.05. We conclude that there is a positive correlation between the variables.

60. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 60 Conclusion We conclude that there is a positive correlation between the number of hours spent watching television and the cholesterol measurement.

61. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 61 Note Even though a significance test indicates the existence of a correlation between x and y in the population, it does not signify a cause-and-effect relationship.