Chapter 10. Correlation and Regression. McGraw-Hill, Bluman, 7th ed., Chapter 10 1

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1 Chapter 10 Correlation and Regression McGraw-Hill, Bluman, 7th ed., Chapter 10 1

2 Chapter 10 Overview Introduction 10-1 Scatter Plots and Correlation 10- Regression 10-3 Coefficient of Determination and Standard Error of the Estimate 10-4 Multiple Regression (Optional) Bluman, Chapter 10

3 Chapter 10 Objectives 1. Draw a scatter plot for a set of ordered pairs.. Compute the correlation coefficient. 3. Test the hypothesis H 0 : ρ = Compute the equation of the regression line. 5. Compute the coefficient of determination. 6. Compute the standard error of the estimate. 7. Find a prediction interval. 8. Be familiar with the concept of multiple regression. Bluman, Chapter 10 3

4 Introduction In addition to hypothesis testing and confidence intervals, inferential statistics involves determining whether a relationship between two or more numerical or quantitative variables exists. Bluman, Chapter 10 4

5 Introduction Correlation is a statistical method used to determine whether a linear relationship between variables exists. Regression is a statistical method used to describe the nature of the relationship between variables that is, positive or negative, linear or nonlinear. Bluman, Chapter 10 5

6 Introduction The purpose of this chapter is to answer these questions statistically: 1. Are two or more variables related?. If so, what is the strength of the relationship? 3. What type of relationship exists? 4. What kind of predictions can be made from the relationship? Bluman, Chapter 10 6

7 Introduction 1. Are two or more variables related?. If so, what is the strength of the relationship? To answer these two questions, statisticians use the correlation coefficient, a numerical measure to determine whether two or more variables are related and to determine the strength of the relationship between or among the variables. Bluman, Chapter 10 7

8 Introduction 3. What type of relationship exists? There are two types of relationships: simple and multiple. In a simple relationship, there are two variables: an independent variable (predictor variable) and a dependent variable (response variable). In a multiple relationship, there are two or more independent variables that are used to predict one dependent variable. Bluman, Chapter 10 8

9 Introduction 4. What kind of predictions can be made from the relationship? Predictions are made in all areas and daily. Examples include weather forecasting, stock market analyses, sales predictions, crop predictions, gasoline price predictions, and sports predictions. Some predictions are more accurate than others, due to the strength of the relationship. That is, the stronger the relationship is between variables, the more accurate the prediction is. Bluman, Chapter 10 9

10 10.1 Scatter Plots and Correlation A scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable x and the dependent variable y. Bluman, Chapter 10 10

11 Chapter 10 Correlation and Regression Section 10-1 Example 10-1 Page #536 Bluman, Chapter 10 11

12 Example 10-1: Car Rental Companies Construct a scatter plot for the data shown for car rental companies in the United States for a recent year. Step 1: Draw and label the x and y axes. Step : Plot each point on the graph. Bluman, Chapter 10 1

13 Example 10-1: Car Rental Companies Positive Relationship Bluman, Chapter 10 13

14 Chapter 10 Correlation and Regression Section 10-1 Example 10- Page #537 Bluman, Chapter 10 14

15 Example 10-: Absences/Final Grades Construct a scatter plot for the data obtained in a study on the number of absences and the final grades of seven randomly selected students from a statistics class. Step 1: Draw and label the x and y axes. Step : Plot each point on the graph. Bluman, Chapter 10 15

16 Example 10-: Absences/Final Grades Negative Relationship Bluman, Chapter 10 16

17 Chapter 10 Correlation and Regression Section 10-1 Example 10-3 Page #538 Bluman, Chapter 10 17

18 Example 10-3: Exercise/Milk Intake Construct a scatter plot for the data obtained in a study on the number of hours that nine people exercise each week and the amount of milk (in ounces) each person consumes per week. Step 1: Draw and label the x and y axes. Step : Plot each point on the graph. Bluman, Chapter 10 18

19 Example 10-3: Exercise/Milk Intake Very Weak Relationship Bluman, Chapter 10 19

20 Correlation The correlation coefficient computed from the sample data measures the strength and direction of a linear relationship between two variables. There are several types of correlation coefficients. The one explained in this section is called the Pearson product moment correlation coefficient (PPMC). The symbol for the sample correlation coefficient is r. The symbol for the population correlation coefficient is ρ. Bluman, Chapter 10 0

21 Correlation The range of the correlation coefficient is from 1 to +1. If there is a strong positive linear relationship between the variables, the value of r will be close to +1. If there is a strong negative linear relationship between the variables, the value of r will be close to 1. Bluman, Chapter 10 1

22 Correlation Bluman, Chapter 10

23 Correlation Coefficient The formula for the correlation coefficient is r = ( ) ( )( ) n xy x y ( ) ( ) ( ) ( ) n x x n y y where n is the number of data pairs. Rounding Rule: Round to three decimal places. Bluman, Chapter 10 3

24 Chapter 10 Correlation and Regression Section 10-1 Example 10-4 Page #540 Bluman, Chapter 10 4

25 Example 10-4: Car Rental Companies Compute the correlation coefficient for the data in Example Company A B C D E F Cars x (in 10,000s) Σ x = Income y (in billions) xy x y Σ y = Σ xy = Σ x = Σ y = Bluman, Chapter 10 5

26 Example 10-4: Car Rental Companies Compute the correlation coefficient for the data in Example Σx = 153.8, Σy = 18.7, Σxy = 68.77, Σx = , Σy = 80.67, n = 6 r r r = = ( ) ( )( ) n xy x y ( ) ( ) ( ) ( ) n x x n y y ( 6)( 68.77) ( 153.8)( 18.7) ( 6)( ) ( 153.8) ( 6)( 80.67) ( 18.7) = 0.98 (strong positive relationship) Bluman, Chapter 10 6

27 Chapter 10 Correlation and Regression Section 10-1 Example 10-5 Page #541 Bluman, Chapter 10 7

28 Example 10-5: Absences/Final Grades Compute the correlation coefficient for the data in Example 10. Student A B C D E F Number of absences, x Final Grade y (pct.) xy x y ,74 7,396 1,849 5,476 3,364 8,100 G ,084 Σ x = 57 Σ y = 511 Σ xy = 3745 Σ x = 579 Σ y = 38,993 Bluman, Chapter 10 8

29 Example 10-5: Absences/Final Grades Compute the correlation coefficient for the data in Example 10. Σx = 57, Σy = 511, Σxy = 3745, Σx = 579, Σy = 38,993, n = 7 r r r = = ( ) ( )( ) n xy x y ( ) ( ) ( ) ( ) n x x n y y ( 7)( 3745) ( 57)( 511) ( 7)( 579) ( 57) ( 7)( 38, 993) ( 511) = (strong negative relationship) Bluman, Chapter 10 9

30 Chapter 10 Correlation and Regression Section 10-1 Example 10-6 Page #54 Bluman, Chapter 10 30

31 Example 10-6: Exercise/Milk Intake Compute the correlation coefficient for the data in Example Subject Hours, x Amount y xy x y A B C D E F , ,04 4, ,04 G ,136 H ,184 I ,304 Σ x = 36 Σ y = 370 Σ xy = 1,50 Σ x = 3 Σ y = 19,36 Bluman, Chapter 10 31

32 Example 10-6: Exercise/Milk Intake Compute the correlation coefficient for the data in Example Σx = 36, Σy = 370, Σxy = 150, Σx = 3, Σy = 19,36, n = 9 r r r = = ( ) ( )( ) n xy x y ( ) ( ) ( ) ( ) n x x n y y ( 7)( 150) ( 36)( 370) ( 7)( 3) ( 36) ( 7)( 19, 36) ( 370) = (very weak relationship) Bluman, Chapter 10 3

33 Hypothesis Testing In hypothesis testing, one of the following is true: H 0 : ρ = 0 This null hypothesis means that there is no correlation between the x and y variables in the population. H 1 : ρ 0 This alternative hypothesis means that there is a significant correlation between the variables in the population. Bluman, Chapter 10 33

34 t Test for the Correlation Coefficient t = r n 1 r with degrees of freedom equal to n. Bluman, Chapter 10 34

35 Chapter 10 Correlation and Regression Section 10-1 Example 10-7 Page #544 Bluman, Chapter 10 35

36 Example 10-7: Car Rental Companies Test the significance of the correlation coefficient found in Example Use α = 0.05 and r = Step 1: State the hypotheses. H 0 : ρ = 0 and H 1 : ρ 0 Step : Find the critical value. Since α = 0.05 and there are 6 = 4 degrees of freedom, the critical values obtained from Table F are ±.776. Bluman, Chapter 10 36

37 Example 10-7: Car Rental Companies Step 3: Compute the test value. n 6 t = r = 0.98 = r 1 ( 0.98) Step 4: Make the decision. Reject the null hypothesis. Step 5: Summarize the results. There is a significant relationship between the number of cars a rental agency owns and its annual income. Bluman, Chapter 10 37

38 Chapter 10 Correlation and Regression Section 10-1 Example 10-8 Page #545 Bluman, Chapter 10 38

39 Example 10-8: Car Rental Companies Using Table I, test the significance of the correlation coefficient r = 0.067, from Example 10 6, at α = Step 1: State the hypotheses. H 0 : ρ = 0 and H 1 : ρ 0 There are 9 = 7 degrees of freedom. The value in Table I when α = 0.01 is For a significant relationship, r must be greater than or less than Since r = 0.067, do not reject the null. Hence, there is not enough evidence to say that there is a significant linear relationship between the variables. Bluman, Chapter 10 39

40 Possible Relationships Between Variables When the null hypothesis has been rejected for a specific a value, any of the following five possibilities can exist. 1. There is a direct cause-and-effect relationship between the variables. That is, x causes y.. There is a reverse cause-and-effect relationship between the variables. That is, y causes x. 3. The relationship between the variables may be caused by a third variable. 4. There may be a complexity of interrelationships among many variables. 5. The relationship may be coincidental. Bluman, Chapter 10 40

41 Possible Relationships Between Variables 1. There is a reverse cause-and-effect relationship between the variables. That is, y causes x. For example, water causes plants to grow poison causes death heat causes ice to melt Bluman, Chapter 10 41

42 Possible Relationships Between Variables. There is a reverse cause-and-effect relationship between the variables. That is, y causes x. For example, Suppose a researcher believes excessive coffee consumption causes nervousness, but the researcher fails to consider that the reverse situation may occur. That is, it may be that an extremely nervous person craves coffee to calm his or her nerves. Bluman, Chapter 10 4

43 Possible Relationships Between Variables 3. The relationship between the variables may be caused by a third variable. For example, If a statistician correlated the number of deaths due to drowning and the number of cans of soft drink consumed daily during the summer, he or she would probably find a significant relationship. However, the soft drink is not necessarily responsible for the deaths, since both variables may be related to heat and humidity. Bluman, Chapter 10 43

44 Possible Relationships Between Variables 4. There may be a complexity of interrelationships among many variables. For example, A researcher may find a significant relationship between students high school grades and college grades. But there probably are many other variables involved, such as IQ, hours of study, influence of parents, motivation, age, and instructors. Bluman, Chapter 10 44

45 Possible Relationships Between Variables 5. The relationship may be coincidental. For example, A researcher may be able to find a significant relationship between the increase in the number of people who are exercising and the increase in the number of people who are committing crimes. But common sense dictates that any relationship between these two values must be due to coincidence. Bluman, Chapter 10 45

46 10. Regression If the value of the correlation coefficient is significant, the next step is to determine the equation of the regression line which is the data s line of best fit. Bluman, Chapter 10 46

47 Regression Best fit means that the sum of the squares of the vertical distance from each point to the line is at a minimum. Bluman, Chapter 10 47

48 Regression Line y = a + bx a = ( )( y x ) ( x)( xy) ( ) ( ) n x x b = where a = y intercept b = the slope of the line. ( ) ( )( ) ( ) ( ) n x x n xy x y Bluman, Chapter 10 48

49 Chapter 10 Correlation and Regression Section 10- Example 10-9 Page #553 Bluman, Chapter 10 49

50 Example 10-9: Car Rental Companies Find the equation of the regression line for the data in Example 10 4, and graph the line on the scatter plot. Σx = 153.8, Σy = 18.7, Σxy = 68.77, Σx = , Σy = 80.67, n = 6 a b = = ( y)( x ) ( x)( xy) n( x ) ( x) = ( ) ( )( ) n( x ) ( x) n xy x y y = a + bx y = x ( 18.7)( ) ( 153.8)( 68.77) 6( ) ( 153.8) = ( ) ( )( ) 6( ) ( 153.8) = = Bluman, Chapter 10 50

51 Example 10-9: Car Rental Companies Find two points to sketch the graph of the regression line. Use any x values between 10 and 60. For example, let x equal 15 and 40. Substitute in the equation and find the corresponding y value. y = x ( ) = = y = x ( ) = Plot (15,1.986) and (40,4.636), and sketch the resulting line. = Bluman, Chapter 10 51

52 Example 10-9: Car Rental Companies Find the equation of the regression line for the data in Example 10 4, and graph the line on the scatter plot. y = x ( 15, 1.986) ( 40, 4.636) Bluman, Chapter 10 5

53 Chapter 10 Correlation and Regression Section 10- Example Page #555 Bluman, Chapter 10 53

54 Example 10-11: Car Rental Companies Use the equation of the regression line to predict the income of a car rental agency that has 00,000 automobiles. x = 0 corresponds to 00,000 automobiles. y = x ( ) = =.516 Hence, when a rental agency has 00,000 automobiles, its revenue will be approximately $.516 billion. Bluman, Chapter 10 54

55 Regression The magnitude of the change in one variable when the other variable changes exactly 1 unit is called a marginal change. The value of slope b of the regression line equation represents the marginal change. For valid predictions, the value of the correlation coefficient must be significant. When r is not significantly different from 0, the best predictor of y is the mean of the data values of y. Bluman, Chapter 10 55

56 Assumptions for Valid Predictions 1. For any specific value of the independent variable x, the value of the dependent variable y must be normally distributed about the regression line. See Figure 10 16(a).. The standard deviation of each of the dependent variables must be the same for each value of the independent variable. See Figure 10 16(b). Bluman, Chapter 10 56

57 Extrapolations (Future Predictions) Extrapolation, or making predictions beyond the bounds of the data, must be interpreted cautiously. Remember that when predictions are made, they are based on present conditions or on the premise that present trends will continue. This assumption may or may not prove true in the future. Bluman, Chapter 10 57

58 Procedure Table Step 1: Make a table with subject, x, y, xy, x, and y columns. Step : Find the values of xy, x, and y. Place them in the appropriate columns and sum each column. Step 3: Substitute in the formula to find the value of r. Step 4: When r is significant, substitute in the formulas to find the values of a and b for the regression line equation y = a + bx. Bluman, Chapter 10 58

59 10.3 Coefficient of Determination and Standard Error of the Estimate ( y) The total variation y is the sum of the squares of the vertical distances each point is from the mean. The total variation can be divided into two parts: that which is attributed to the relationship of x and y, and that which is due to chance. Bluman, Chapter 10 59

60 Variation The variation obtained from the relationship (i.e., from the predicted y' values) is ( y y) and is called the explained variation. Variation due to chance, found by ( y y), is called the unexplained variation. This variation cannot be attributed to the relationships. Bluman, Chapter 10 60

61 Variation Bluman, Chapter 10 61

62 Coefficient of Determiation The coefficient of determination is the ratio of the explained variation to the total variation. The symbol for the coefficient of determination is r. explained variation r = total variation Another way to arrive at the value for r is to square the correlation coefficient. Bluman, Chapter 10 6

63 Coefficient of Nondetermiation The coefficient of nondetermination is a measure of the unexplained variation. The formula for the coefficient of determination is 1.00 r. Bluman, Chapter 10 63

64 Standard Error of the Estimate The standard error of estimate, denoted by s est is the standard deviation of the observed y values about the predicted y' values. The formula for the standard error of estimate is: s est = ( y y ) n Bluman, Chapter 10 64

65 Chapter 10 Correlation and Regression Section 10-3 Example 10-1 Page #569 Bluman, Chapter 10 65

66 Example 10-1: Copy Machine Costs A researcher collects the following data and determines that there is a significant relationship between the age of a copy machine and its monthly maintenance cost. The regression equation is y = x. Find the standard error of the estimate. Bluman, Chapter 10 66

67 Example 10-1: Copy Machine Costs Machine A B C D E F Age x (years) ( ) ( ) ( ) ( ) ( ) Monthly cost, y y y y (y y ) y = x y = = y = = y = = y = = y = = s est s est = ( y y ) n = = Bluman, Chapter 10 67

68 Chapter 10 Correlation and Regression Section 10-3 Example Page #570 Bluman, Chapter 10 68

69 Example 10-13: Copy Machine Costs s est = y a y b xy n Bluman, Chapter 10 69

70 Example 10-13: Copy Machine Costs Machine A B C D E F Age x (years) Monthly cost, y xy y ,844 6,084 4,900 8,100 8,649 10,609 4,186 s est s est = y a y b xy n ( ) ( ) 4, = = Bluman, Chapter 10 70

71 Formula for the Prediction Interval about a Value y ( X) 1 n x y tα sest 1+ + < y n n x x ( ) ( X) 1 n x < y + t s 1+ + n n x x α est ( ) with d.f. = n - Bluman, Chapter 10 71

72 Chapter 10 Correlation and Regression Section 10-3 Example Page #571 Bluman, Chapter 10 7

73 Example 10-14: Copy Machine Costs For the data in Example 10 1, find the 95% prediction interval for the monthly maintenance cost of a machine that is 3 years old. Step 1: Find x, x, and X. 0 x= 0 x = 8 X = = Step : Find y for x = 3. y = ( 3) = Step 3: Find s est. s est = 6.48 (as shown in Example 10-13) Bluman, Chapter 10 73

74 Example 10-14: Copy Machine Costs Step 4: Substitute in the formula and solve. ( X) ( ) 1 n x y tα sest 1+ + < y n n x x ( X) ( ) 1 n x < y + t s 1+ + n n x x α est 6( 3 3.3) ( ) ( ) (.776)( 6.48) 1+ + < y ( 3 3.3) ( ) ( ) 1 < (.776)( 6.48) Bluman, Chapter 10 74

75 Example 10-14: Copy Machine Costs Step 4: Substitute in the formula and solve. 6( 3 3.3) ( ) ( ) (.776)( 6.48) 1+ + < ( 3 3.3) ( ) ( ) 1 < (.776)( 6.48) < y < < y < Hence, you can be 95% confident that the interval < y < contains the actual value of y. y Bluman, Chapter 10 75

76 10.4 Multiple Regression (Optional) In multiple regression, there are several independent variables and one dependent variable, and the equation is y = a+ bx + bx + + b x 1 1 k k where x, x,, x = independent variables. 1 k Bluman, Chapter 10 76

77 Assumptions for Multiple Regression 1. normality assumption for any specific value of the independent variable, the values of the y variable are normally distributed.. equal-variance assumption the variances (or standard deviations) for the y variables are the same for each value of the independent variable. 3. linearity assumption there is a linear relationship between the dependent variable and the independent variables. 4. nonmulticollinearity assumption the independent variables are not correlated. 5. independence assumption the values for the y variables are independent. Bluman, Chapter 10 77

78 Multiple Correlation Coefficient In multiple regression, as in simple regression, the strength of the relationship between the independent variables and the dependent variable is measured by a correlation coefficient. This multiple correlation coefficient is symbolized by R. Bluman, Chapter 10 78

79 Multiple Correlation Coefficient The formula for R is R where = + yx yx yx yx x x r r r r r r xx 1 r yx1 = correlation coefficient for y and x 1 r yx = correlation coefficient for y and x r x1 x = correlation coefficient for x 1 and x Bluman, Chapter 10 79

80 Chapter 10 Correlation and Regression Section 10-4 Example Page #576 Bluman, Chapter 10 80

81 Example 10-15: State Board Scores A nursing instructor wishes to see whether a student s grade point average and age are related to the student s score on the state board nursing examination. She selects five students and obtains the following data. Find the value of R. Bluman, Chapter 10 81

82 Example 10-15: State Board Scores A nursing instructor wishes to see whether a student s grade point average and age are related to the student s score on the state board nursing examination. She selects five students and obtains the following data. Find the value of R. The values of the correlation coefficients are r xx r r 1 yx yx 1 = = = Bluman, Chapter 10 8

83 Example 10-15: State Board Scores R = R = R = r + r r r r yx yx yx yx x x r xx 1 ( 0.845) + ( 0.791) ( 0.845)( 0.791)( 0.371) 1 ( 0.371) Hence, the correlation between a student s grade point average and age with the student s score on the nursing state board examination is In this case, there is a strong relationship among the variables; the value of R is close to Bluman, Chapter 10 83

84 F Test for Significance of R The formula for the F test is where F = ( 1 R ) ( n k 1) n = the number of data groups k = the number of independent variables. d.f.n. = n k d.f.d. = n k 1 R k Bluman, Chapter 10 84

85 Chapter 10 Correlation and Regression Section 10-4 Example Page #577 Bluman, Chapter 10 85

86 Example 10-16: State Board Scores Test the significance of the R obtained in Example at α = F = F = R ( 1 R ) ( n k 1) The critical value obtained from Table H with a 0.05, d.f.n. = 3, and d.f.d. = is Hence, the decision is to reject the null hypothesis and conclude that there is a significant relationship among the student s GPA, age, and score on the nursing state board examination. k = ( ) ( ) Bluman, Chapter 10 86

87 Adjusted R The formula for the adjusted R is R adj = 1 ( 1 R )( n 1) n k 1 Bluman, Chapter 10 87

88 Chapter 10 Correlation and Regression Section 10-4 Example Page #578 Bluman, Chapter 10 88

89 Example 10-17: State Board Scores Calculate the adjusted R for the data in Example The value for R is R adj = 1 ( 1 R )( n 1) n k 1 R adj = 1 ( )( 5 1) 5 1 = In this case, when the number of data pairs and the number of independent variables are accounted for, the adjusted multiple coefficient of determination is Bluman, Chapter 10 89