Chapter 9. Chapter 9. Correlation and Regression. Overview. Correlation and Regression. Paired Data Overview 9-22 Correlation 9-33 Regression

Transcription

1 Chapter 9 Correlation and Regression 1 Chapter 9 Correlation and Regression 9-11 Overview 9-22 Correlation 9-33 Regression 2 Overview Paired Data is there a relationship if so, what is the equation use the equation for prediction 3

2 Eample: Lengths and Weights of Male Bears Length (in.) Weight (lb) Correlation 5 Definition Correlation eists between two variables when one of them is related to the other in some wa 6

3 Definition Scatterplot (or scatter diagram) is a graph in which the paired (,)) sample data are plotted with a horizontal ais and a vertical ais. Each individual (,(,) ) pair is plotted as a single point. 7 Eample: Lengths and Weights of Male Bears Length (in.) Weight (lb) (, ) = (Length, Weight) (53.0, 80) (67.5, 344) (72.0, 416) etc. 8 Scatter Diagram of Paired Data Lengths and Weights of Male Bears 500 Weight (lb.) (72,416) (68.5,360) (72,348) (67.5,344) (73,332) (73.5,262) (37,34) (53,80) Length (in.) 9

4 Scatter Diagram of Paired Data Lengths and Weights of Male Bears 500 Weight (lb.) Length (in.) 10 Positive Linear Correlation (a) Positive (b) Strong positive (c) Perfect positive Figure 9-29 Scatter Plots 11 Negative Linear Correlation (d) Negative (e) Strong negative (f) Perfect negative Figure 9-29 Scatter Plots 12

5 No Linear Correlation (g) No Correlation (h) Nonlinear Correlation Figure 9-29 Scatter Plots 13 Definition Linear Correlation Coefficient r measures strength of the linear relationship between paired - and -quantitative values in a sample 14 Definition Linear Correlation Coefficient r sometimes referred to as the Pearson product moment correlation coefficient 15

6 Assumptions 1. The sample of paired data (,(,) ) is a random sample. 2. The pairs of (,(,) ) data have a bivariate normal distribution. 16 n Σ Σ Σ 2 (Σ) 2 Σ r ρ Notation for the Linear Correlation Coefficient number of pairs of data presented. denotes the addition of the items indicated. denotes the sum of all values. indicates that each score should be squared and then those squares added. indicates that the scores should be added and the total then squared. indicates that each score should be first multiplied b its corresponding score. After obtaining all such products, find their sum. represents linear correlation coefficient for a sample represents linear correlation coefficient for a population 17 Definition Linear Correlation Coefficient r r = nσ - (Σ)(Σ) n(σ 2 ) - (Σ) 2 n(σ 2 ) - (Σ) 2 Formula 9-19 Calculators can compute r ρ (rho)) is the linear correlation coefficient for all paired data in the population. 18

7 Rounding the Linear Correlation Coefficient r Round to three decimal places so that it can be compared to critical values in Table A-5A Use calculator or computer if possible 19 Eample: Lengths and Weights of Male Bears Length (in.) Weight (lb) r = using our calculator 20 Interpreting the Linear Correlation Coefficient If the absolute value of r eceeds the value in Table A - 5, conclude that there is a significant linear correlation. Otherwise, there is not sufficient evidence to support the conclusion of significant linear correlation. 21

8 TABLE A-5 A Critical Values of the Pearson Correlation Coefficient r n α =.05 α = Properties of the Linear Correlation Coefficient r r 1 2. Value of r does not change if all values of either variable are converted to a different scale. 3. The value of r is not affected b the choice of and.. Interchange and and the value of r will not change. 4. r measures strength of a linear relationship. 23 Common Errors Involving Correlation 1. Causation: It is incorrect to conclude that correlation implies causalit. 2. Averages: : Averages suppress individual variation and ma inflate the correlation coefficient. 3. Linearit: There ma be some relationship between and even when there is no significant linear correlation. 24

9 Common Errors Involving Correlation Distance (feet) Time (seconds) Scatterplot of Distance above Ground and Time for Object Thrown Upward 25 Formal Hpothesis Test To determine whether there is a significant linear correlation between two variables Two methods Both methods let H 0 : ρ = 0 (no significant linear correlation) H 1 : ρ 0 (significant linear correlation) 26 Method 1: Test Statistic is t (follows format of earlier chapters) Test statistic: r t = 1 - r 2 n -2 Critical values: use Table A-3 A 3 with degrees of freedom = n

10 Method 1: Test Statistic is t (follows format of earlier chapters) 28 Method 2: Test Statistic is r (uses fewer calculations) Test statistic: r Critical values: Refer to Table A-5 A (no degrees of freedom) Reject ρ = 0 Fail to reject ρ = 0 Reject ρ = 0-1 r = r = Sample data: r = Testing for a Linear Correlation A-5 30

11 Is there a significant linear correlation? Data from the Garbage Project Plastic (lb) Household n = 8 α = 0.05 H 0 : ρ = 0 H 1 :ρ 0 Test statistic is r = Is there a significant linear correlation? n = 8 α = 0.05 H 0 : ρ = 0 H 1 :ρ 0 Test statistic is r = Critical values are r = and (Table A-5 with n = 8 and α = 0.05) TABLE A-5 A 5 Critical Values of the Pearson Correlation Coefficient r n α =.05 α = Is there a significant linear correlation? > The test statistic does fall within the critical region. Therefore, we REJECT H 0 : ρ = 0 (no correlation) and conclude there is a significant linear correlation between the weights of discarded plastic and household size. Reject ρ = 0 Fail to reject ρ = 0 Reject ρ = 0-1 r = r = Sample data: r =

12 Formula 9-19 is developed from r = Justification for r Formula Σ ( -) ( -) (n -1) S S (, ) centroid of sample points = 3 - = 7-3 = 4 (7, 23) - = = Quadrant 2 Quadrant 1 Quadrant 3 (, ) Quadrant 4 =