1. 1 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Chapter 9 Correlation and Regression

2. 2 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Chapter 9 Correlation and Regression 9-1 Overview 9-2 Correlation 9-3 Regression 9-4 Variation and Prediction Intervals 9-5 Multiple Regression 9-6 Modeling

3. 3 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 9-1 Overview Paired Data  is there a relationship  if so, what is the equation  use the equation for prediction

4. 4 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 9-2 Correlation

5. 5 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition  Correlation exists between two variables when one of them is related to the other in some way

6. 6 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Assumptions 1. The sample of paired data ( x,y ) is a random sample. 2. The pairs of ( x,y ) data have a bivariate normal distribution.

7. 7 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition  Scatterplot (or scatter diagram) is a graph in which the paired ( x,y ) sample data are plotted with a horizontal x axis and a vertical y axis. Each individual ( x,y ) pair is plotted as a single point.

8. 8 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Scatter Diagram of Paired Data

9. 9 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Positive Linear Correlation x x y yy x Figure 9-1 Scatter Plots (a) Positive (b) Strong positive (c) Perfect positive

10. 10 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Negative Linear Correlation x x y yy x (d) Negative (e) Strong negative (f) Perfect negative Figure 9-1 Scatter Plots

11. 11 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman No Linear Correlation x x y y (g) No Correlation (h) Nonlinear Correlation Figure 9-1 Scatter Plots

12. 12 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition  Linear Correlation Coefficient r measures strength of the linear relationship between paired x and y values in a sample

13. 13 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman n  xy - (  x)(  y) n(  x 2 ) - (  x) 2 n(  y 2 ) - (  y) 2 r = Definition  Linear Correlation Coefficient r Formula 9-1 measures strength of the linear relationship between paired x and y values in a sample

14. 14 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman n  xy - (  x)(  y) n(  x 2 ) - (  x) 2 n(  y 2 ) - (  y) 2 r = Definition  Linear Correlation Coefficient r Formula 9-1 measures strength of the linear relationship between paired x and y values in a sample Calculators can compute r  (rho) is the linear correlation coefficient for all paired data in the population.

15. 15 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Notation for the Linear Correlation Coefficient n = number of pairs of data presented  denotes the addition of the items indicated.  x denotes the sum of all x values.  x 2 indicates that each x score should be squared and then those squares added. (  x ) 2 indicates that the x scores should be added and the total then squared.  xy indicates that each x score should be first multiplied by its corresponding y score. After obtaining all such products, find their sum. r represents linear correlation coefficient for a sample  represents linear correlation coefficient for a population

16. 16 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman  Round to three decimal places so that it can be compared to critical values in Table A-6  Use calculator or computer if possible Rounding the Linear Correlation Coefficient r

17. 17 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Interpreting the Linear Correlation Coefficient  If the absolute value of r exceeds the value in Table A - 6, conclude that there is a significant linear correlation.  Otherwise, there is not sufficient evidence to support the conclusion of significant linear correlation.

18. 18 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman TABLE A-6 Critical Values of the Pearson Correlation Coefficient r 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 35 40 45 50 60 70 80 90 100 n.999.959.917.875.834.798.765.735.708.684.661.641.623.606.590.575.561.505.463.430.402.378.361.330.305.286.269.256.950.878.811.754.707.666.632.602.576.553.532.514.497.482.468.456.444.396.361.335.312.294.279.254.236.220.207.196   =.05   =.01

19. 19 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Properties of the Linear Correlation Coefficient r 1. -1  r  1 2. Value of r does not change if all values of either variable are converted to a different scale. 3. The r is not affected by the choice of x and y. Interchange x and y and the value of r will not change. 4. r measures strength of a linear relationship.

20. 20 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Common Errors Involving Correlation 1. Causation: It is wrong to conclude that correlation implies causality. 2. Averages: Averages suppress individual variation and may inflate the correlation coefficient. 3. Linearity: There may be some relationship between x and y even when there is no significant linear correlation.

21. 21 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 50 100 150 200 250 0 12345678 Distance (feet) Time (seconds) FIGURE 9-2 Common Errors Involving Correlation Scatterplot of Distance above Ground and Time for Object Thrown Upward

22. 22 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Formal Hypothesis Test  To determine whether there is a significant linear correlation between two variables  Two methods  Both methods letH 0 :  =  (no significant linear correlation) H 1 :    (significant linear correlation)

23. 23 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Method 1: Test Statistic is t (follows format of earlier chapters) Test statistic: 1 - r 2 n - 2 r t =

24. 24 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Method 1: Test Statistic is t (follows format of earlier chapters) Test statistic: 1 - r 2 n - 2 r Critical values: use Table A-3 with degrees of freedom = n - 2 t =

25. 25 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Figure 9-4 Method 1: Test Statistic is t (follows format of earlier chapters)

26. 26 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman  Test statistic: r  Critical values: Refer to Table A-6 (no degrees of freedom) Method 2: Test Statistic is r (uses fewer calculations)

27. 27 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman  Test statistic: r  Critical values: Refer to Table A-6 (no degrees of freedom) Method 2: Test Statistic is r (uses fewer calculations) Fail to reject  = 0 0 r = - 0.811 r = 0.811 1 Sample data: r = 0.828 Figure 9-5 Reject  = 0 Reject  = 0

28. 28 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman FIGURE 9-3 Testing for a Linear Correlation If H 0 is rejected conclude that there is a significant linear correlation. If you fail to reject H 0, then there is not sufficient evidence to conclude that there is linear correlation. If the absolute value of the test statistic exceeds the critical values, reject H 0 :  = 0 Otherwise fail to reject H 0 The test statistic is t = 1 - r 2 n -2 r Critical values of t are from Table A-3 with n -2 degrees of freedom The test statistic is Critical values of t are from Table A-6 r Calculate r using Formula 9-1 Select a significance level  Let H 0 :  = 0 H 1 :   0 Start METHOD 1 METHOD 2

29. 29 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0.27 2 1.41 3 2.19 3 2.83 6 2.19 4 1.81 2 0.85 1 3.05 5 Data from the Garbage Project x Plastic (lb) y Household Is there a significant linear correlation?

30. 30 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0.27 2 1.41 3 2.19 3 2.83 6 2.19 4 1.81 2 0.85 1 3.05 5 Data from the Garbage Project x Plastic (lb) y Household n = 8  = 0.05 H 0 :  = 0 H 1 :   0 Test statistic is r = 0.842 Is there a significant linear correlation?

31. 31 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman n = 8  = 0.05 H 0 :  = 0 H 1 :   0 Test statistic is r = 0.842 Critical values are r = - 0.707 and 0.707 (Table A-6 with n = 8 and  = 0.05) TABLE A-6 Critical Values of the Pearson Correlation Coefficient r 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 35 40 45 50 60 70 80 90 100 n.999.959.917.875.834.798.765.735.708.684.661.641.623.606.590.575.561.505.463.430.402.378.361.330.305.286.269.256.950.878.811.754.707.666.632.602.576.553.532.514.497.482.468.456.444.396.361.335.312.294.279.254.236.220.207.196   =.05   =.01 Is there a significant linear correlation?

32. 32 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 r = - 0.707 r = 0.707 1 Sample data: r = 0.842 - 1 Is there a significant linear correlation? Fail to reject  = 0 Reject  = 0 Reject  = 0

33. 33 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 r = - 0.707 r = 0.707 1 Sample data: r = 0.842 - 1 0.842 > 0.707, That is the test statistic does fall within the critical region. Is there a significant linear correlation? Fail to reject  = 0 Reject  = 0 Reject  = 0

34. 34 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 r = - 0.707 r = 0.707 1 Sample data: r = 0.842 - 1 0.842 > 0.707, That is the test statistic does fall within the critical region. Therefore, we REJECT H 0 :  = 0 (no correlation) and conclude there is a significant linear correlation between the weights of discarded plastic and household size. Is there a significant linear correlation? Fail to reject  = 0 Reject  = 0 Reject  = 0

35. 35 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman r =r =  (x -x) (y -y) (n -1 ) S x S y Justification for r Formula Formula 9-1 is developed from

36. 36 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Justification for r Formula r =r =  (x -x) (y -y) (n -1 ) S x S y (x, y) centroid of sample points Formula 9-1 is developed from

37. 37 Chapter 9. Section 9-1 and 9-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman x = 3 Quadrant 3 Quadrant 2 Quadrant 1 Quadrant 4 x y y = 11 (x, y) x - x = 7- 3 = 4 y - y = 23 - 11 = 12 01234567 0 4 8 12 16 20 24 r =r =  (x -x) (y -y) (n -1 ) S x S y (x, y) centroid of sample points (7, 23) FIGURE 9-6 Formula 9-1 is developed from Justification for r Formula