ELEMENTARY STATISTICS Chapter 9 Correlation and Regression C

Chapter 9 Correlation and Regression 9 -1 Overview 9 -2 Correlation 9 -3 Regression

9 -1 Overview Paired Data v is there a relationship v if so, what

9 -2 Correlation Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics,

Definition v. Correlation exists between two variables when one of them is related to

Assumptions 1. The sample of paired data (x, y) is a random sample. 2.

Definition v. Scatterplot (or scatter diagram) is a graph in which the paired (x,

Scatter Diagram of Paired Data Chapter 9. Section 9 -1 and 9 -2. Triola,

Positive Linear Correlation y y y (a) Positive Figure 9 -1 x x x

Negative Linear Correlation y y y (d) Negative Figure 9 -1 x x x

No Linear Correlation y y x (g) No Correlation Figure 9 -1 x (h)

Definition v. Linear Correlation Coefficient r measures strength of the linear relationship between paired

Notation for the Linear Correlation Coefficient n = number of pairs of data presented

Rounding the Linear Correlation Coefficient r v Round to three decimal places so that

Interpreting the Linear Correlation Coefficient v. If the absolute value of r exceeds the

TABLE A-6 Critical Values of the Pearson Correlation Coefficient r n 4 5 6

Example 1 • Construct a scatter plot for the given data Age, x 43

Example 1 Solution: Draw and label the x and y axes. Plot each point

Example 2 • A Statistics professor at a state university wants to see how

Subject Test Score GPA x y xy x^2 y^2 1 98 2. 1 205.

Example 2 • • Solve of SSxy, SSxx, and Ssyy; SSxy = ∑xy –

Example 2 • Substitute in the formula and solve for r; • r =

Properties of the Linear Correlation Coefficient r 1. -1 r 1 2. Value of

Common Errors Involving Correlation 1. Causation: It is wrong to conclude that correlation implies

Common Errors Involving Correlation FIGURE 9 -2 250 Distance (feet) 200 150 100 50

Formal Hypothesis Test v To determine whethere is a significant linear correlation between two

Method 1: Test Statistic is t (follows format of earlier chapters) Test statistic: t=

Method 1: Test Statistic is t (follows format of earlier chapters) Figure 9 -4

Method 2: Test Statistic is r (uses fewer calculations) v. Test statistic: r v.

FIGURE 9 -3 Start Testing for a Linear Correlation Let H 0: = 0

Is there a significant linear correlation? Data from the Garbage Project x Plastic (lb)

Is there a significant linear correlation? n=8 = 0. 05 =0 : 0 H

Is there a significant linear correlation? Reject = 0 -1 r = - 0.

Is there a significant linear correlation? 0. 842 > 0. 707, That is the

Justification for r Formula Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary

Justification for r Formula 9 -1 is developed from r= (x -x) (y -y)

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

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

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

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. Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6

Definition v. 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. Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 7

Positive Linear Correlation y y y (a) Positive Figure 9 -1 x x x (b) Strong positive (c) Perfect positive Scatter Plots Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 10

Negative Linear Correlation y y y (d) Negative Figure 9 -1 x x x (e) Strong negative (f) Perfect negative Scatter Plots Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 11

No Linear Correlation y y x (g) No Correlation Figure 9 -1 x (h) Nonlinear Correlation Scatter Plots Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 12

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

Definition v. Linear Correlation Coefficient r measures strength of the linear relationship between paired x and y values in a sample r= n xy - ( x)( y) n( x 2) - ( x)2 n( y 2) - ( y)2 Formula 9 -1 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 14

Definition v. Linear Correlation Coefficient r measures strength of the linear relationship between paired x and y values in a sample r= n xy - ( x)( y) n( x 2) - ( x)2 n( y 2) - ( y)2 Formula 9 -1 Calculators can compute r (rho) is the linear correlation coefficient for all paired data in the population. Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 15

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 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 16

Rounding the Linear Correlation Coefficient r v Round to three decimal places so that it can be compared to critical values in Table A-6 v Use calculator or computer if possible Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 17

Interpreting the Linear Correlation Coefficient v. If the absolute value of r exceeds the value in Table A - 6, conclude that there is a significant linear correlation. v. Otherwise, there is not sufficient evidence to support the conclusion of significant linear correlation. Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 18

TABLE A-6 Critical Values of the Pearson Correlation Coefficient r n 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 =. 05. 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 =. 01. 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 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 19

Example 1 • Construct a scatter plot for the given data Age, x 43 48 56 61 67 70 Pressure, y 128 120 135 143 141 152 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 20

Example 1 Solution: Draw and label the x and y axes. Plot each point on the graph below Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 21

Example 2 • A Statistics professor at a state university wants to see how strong the relationship is between a student’s score on a test and his or her grade point average. The data obtained from the sample follow: Test score, x 98 105 100 106 95 116 112 GPA, y 2. 1 2. 4 3. 2 2. 7 2. 2 2. 3 3. 8 3. 4 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 22

Subject Test Score GPA x y xy x^2 y^2 1 98 2. 1 205. 8 9604 4. 41 2 105 2. 4 252 11025 5. 76 3 100 3. 2 320 10000 10. 24 4 100 2. 7 270 10000 7. 29 5 106 2. 2 233. 2 11236 4. 84 6 95 2. 3 218. 5 9025 5. 29 7 116 3. 8 440. 8 13456 14. 44 8 112 3. 4 380. 8 12544 11. 56 SUM 832 22. 1 2321. 1 86890 63. 83 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 23

Example 2 • • Solve of SSxy, SSxx, and Ssyy; SSxy = ∑xy – [(∑x) (∑y )]/n = 2321. 1 – [(832)(22. 1)]/8 = 22. 7 SSxx = ∑x 2 – (∑x)2/n = 86890 – [(832)2]/8 = 362 SSyy = ∑y 2 – (∑y)2/n = 63. 83 – [(22. 1)2]/8 = 2. 78 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 24

Example 2 • Substitute in the formula and solve for r; • r = SSxy/(SSxx * Ssyy)0. 5 • = 22. 7/[(362)(2. 78)]0. 5 = 0. 716 • The correlation coefficient suggests a strong positive relationship between the test score and the grade point average. Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 25

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. Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 26

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. Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 27

Common Errors Involving Correlation FIGURE 9 -2 250 Distance (feet) 200 150 100 50 0 0 1 2 3 4 5 6 7 8 Time (seconds) Scatterplot of Distance above Ground and Time for Object Thrown Upward Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 28

Formal Hypothesis Test v To determine whethere is a significant linear correlation between two variables v Two methods v Both methods let H 0: = (no significant linear correlation) H 1: (significant linear correlation) Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 29

Method 1: Test Statistic is t (follows format of earlier chapters) Test statistic: t= r 1 -r 2 n-2 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 30

Method 1: Test Statistic is t (follows format of earlier chapters) Test statistic: t= r 1 -r 2 n-2 Critical values: use Table A-3 with degrees of freedom = n - 2 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 31

Method 1: Test Statistic is t (follows format of earlier chapters) Figure 9 -4 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 32

Method 2: Test Statistic is r (uses fewer calculations) v. Test statistic: r v. Critical values: Refer to Table A-6 (no degrees of freedom) Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 33

Method 2: Test Statistic is r (uses fewer calculations) v. Test statistic: r v. Critical values: Refer to Table A-6 (no degrees of freedom) Reject = 0 -1 Figure 9 -5 r = - 0. 811 Fail to reject =0 0 Reject = 0 r = 0. 811 1 Sample data: r = 0. 828 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 34

FIGURE 9 -3 Start Testing for a Linear Correlation Let H 0: = 0 H 1: 0 Select a significance level Calculate r using Formula 9 -1 METHOD 2 The test statistic is r The test statistic is t= r Critical values of t are from Table A-6 1 -r 2 n -2 Critical values of t are from Table A-3 with n -2 degrees of freedom If the absolute value of the test statistic exceeds the critical values, reject H 0: = 0 Otherwise fail to reject H 0 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. Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 35

Is there a significant linear correlation? Data from the Garbage Project x Plastic (lb) y Household 0. 27 1. 41 2 3 2. 19 2. 83 2. 19 1. 81 0. 85 3. 05 3 6 4 2 1 5 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 36

Is there a significant linear correlation? Data from the Garbage Project x Plastic (lb) y Household n=8 0. 27 1. 41 2 3 = 0. 05 2. 19 2. 83 2. 19 1. 81 0. 85 3. 05 3 6 4 2 1 5 H 0: = 0 H 1 : 0 Test statistic is r = 0. 842 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 37

Is there a significant linear correlation? n=8 = 0. 05 =0 : 0 H 0 : H 1 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 n 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 =. 05. 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 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman =. 01. 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 38

Is there a significant linear correlation? Reject = 0 -1 r = - 0. 707 Fail to reject =0 0 Reject = 0 r = 0. 707 1 Sample data: r = 0. 842 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 39

Is there a significant linear correlation? 0. 842 > 0. 707, That is the test statistic does fall within the critical region. Reject = 0 -1 r = - 0. 707 Fail to reject =0 0 Reject = 0 r = 0. 707 1 Sample data: r = 0. 842 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 40

Is there a significant linear correlation? 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. Reject = 0 -1 r = - 0. 707 Fail to reject =0 0 Reject = 0 r = 0. 707 1 Sample data: r = 0. 842 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 41

Justification for r Formula 9 -1 is developed from r= (x -x) (y -y) (n -1) Sx Sy Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 43

Justification for r Formula 9 -1 is developed from r= (x -x) (y -y) (n -1) Sx Sy (x, y) centroid of sample points Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 44

Justification for r Formula 9 -1 is developed from r= (x -x) (y -y) (n -1) Sx Sy (x, y) centroid of sample points x=3 y x - x = 7 - 3 = 4 (7, 23) • 24 20 y - y = 23 - 11 = 12 Quadrant 1 Quadrant 2 16 • 12 8 • Quadrant 3 • • 4 y = 11 (x, y) Quadrant 4 FIGURE 9 -6 x 0 0 1 2 3 4 5 6 7 Chapter 9. Section 9 -1 and 9 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 45