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1 9-2 / 9.3 Correlation and Regression

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2 n xy - ( x)( y) n( x 2 ) - ( x) 2 n( y 2 ) - ( y) 2 r = Definition Linear Correlation Coefficient r measures strength of the linear relationship between paired x and y values in a sample

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3 Formula for b 0 and b 1 b 0 = (y-intercept) ( y) ( x 2 ) - ( x) ( xy) n( xy) - ( x) ( y) n( x 2 ) - ( x) 2 b 1 = (slope) n( x 2 ) - ( x) 2

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Data from the Garbage Project x Plastic (lb) y Household Review Calculations Find the Correlation and the Regression Equation (Line of Best Fit)

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Data from the Garbage Project x Plastic (lb) y Household b 0 = b 1 = 1.48 Using a calculator: y = x r = Review Calculations

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6 Notes on correlation r represents linear correlation coefficient for a sample (ro) represents linear correlation coefficient for a population -1 r 1 r measures strength of a linear relationship. -1 is perfect negative correlation & 1 is perfect positive correlation

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7 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.

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8 Formal Hypothesis Test Two methods Both methods letH 0 : = (no significant linear correlation) H 1 : (significant linear correlation)

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9 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 =

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10 Test statistic: r Critical values: Refer to Table A-6 (no degrees of freedom) Much easier Method 2: Test Statistic is r (uses fewer calculations)

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11 TABLE A-6 Critical Values of the Pearson Correlation Coefficient r n =.05 =.01

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Data from the Garbage Project x Plastic (lb) y Household n = 8 = 0.05 H 0 : = 0 H 1 : 0 Test statistic is r = Is there a significant linear correlation?

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13 n = 8 = 0.05 H 0 : = 0 H 1 : 0 Test statistic is r = Critical values are r = and (Table A-6 with n = 8 and = 0.05) TABLE A-6 Critical Values of the Pearson Correlation Coefficient r n =.05 =.01 Is there a significant linear correlation?

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14 0 r = r = Sample data: r = > 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

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15 0 r = r = Sample data: r = > 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

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16 Regression Definition Regression Model Regression Equation y = b 0 + b 1 x ^ Given a collection of paired data, the regression equation algebraically describes the relationship between the two variables y = 0 + 1 x +

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17 Notation for Regression Equation y -intercept of regression equation 0 b 0 Slope of regression equation 1 b 1 Equation of the regression line y = 0 + 1 x + y = b 0 + b 1 Population Parameter Sample Statistic x ^

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18 Regression Definition Regression Equation Given a collection of paired data, the regression equation Regression Line (line of best fit or least-squares line) is the graph of the regression equation y = b 0 + b 1 x ^ algebraically describes the relationship between the two variables

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19 Assumptions & Observations 1. We are investigating only linear relationships. 2. For each x value, y is a random variable having a normal distribution. 3. There are many methods for determining normality. 3. The regression line goes through (x, y)

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20 1. If there is no significant linear correlation, don’t use the regression equation to make predictions. 2. Stay within the scope of the available sample data when making prediction. Guidelines for Using The Regression Equation

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21 Definitions Outlier a point lying far away from the other data points Influential Points points which strongly affect the graph of the regression line

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22 Definitions Residual (error) for a sample of paired ( x,y ) data, the difference ( y - y ) between an observed sample y -value and the value of y, which is the value of y that is predicted by using the regression equation. Least-Squares Property A straight line satisfies this property if the sum of the squares of the residuals is the smallest sum possible. ^ Residuals and the Least-Squares Property ^

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23 Residuals and the Least-Squares Property x y y = x x y Residual = 7 Residual = -13 Residual = -5 Residual = 11 ^

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24 Definitions Total Deviation from the mean of the particular point ( x, y ) the vertical distance y - y, which is the distance between the point ( x, y ) and the horizontal line passing through the sample mean y Explained Deviation the vertical distance y - y, which is the distance between the predicted y value and the horizontal line passing through the sample mean y Unexplained Deviation the vertical distance y - y, which is the vertical distance between the point ( x, y ) and the regression line. (The distance y - y is also called a residual, as defined in Section 9-3.) ^ ^ ^

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25 Total deviation ( y - y ) Unexplained deviation ( y - y ) Explained deviation ( y - y ) (5, 32) (5, 25) (5, 17) y = x ^ y = 17 ^ ^ y x Unexplained, Explained, and Total Deviation

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26 ( y - y ) = ( y - y ) + (y - y ) (total deviation) = (explained deviation) + (unexplained deviation) (total variation) = (explained variation) + (unexplained variation) Σ ( y - y ) 2 = Σ ( y - y ) 2 + Σ (y - y) 2 ^ ^ ^ ^

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