1. Copyright ©2011 Nelson Education Limited Linear Regression and Correlation CHAPTER 12

2. Copyright ©2011 Nelson Education LimitedExample The table shows the math achievement test scores for a random sample of n = 10 college freshmen, along with their final calculus grades. Student12345678910 Math test, x39432164574728753452 Calculus grade, y65785282928973985675

3. Copyright ©2011 Nelson Education Limited Example

4. The ANOVA Table SourcedfSSMSF Regression1SSRSSR/(1)MSR/MSE Errorn - 2SSESSE/(n-2) Totaln -1Total SS

5. Copyright ©2011 Nelson Education Limited The Calculus Example: SourcedfSSMSF Regression11449.9741 19.14 Error8606.025975.7532 Total92056.0000

6. Copyright ©2011 Nelson Education Limited Regression Analysis: y versus x The regression equation is y = 40.8 + 0.766 x Predictor Coef SE Coef T P Constant 40.784 8.507 4.79 0.001 x 0.7656 0.1750 4.38 0.002 S = 8.70363 R-Sq = 70.5% R-Sq(adj) = 66.8% Analysis of Variance Source DF SS MS F P Regression 1 1450.0 1450.0 19.14 0.002 Residual Error 8 606.0 75.8 Total 9 2056.0 Regression coefficients, a and b Minitab Output Least squares regression line

7. Copyright ©2011 Nelson Education Limited Measuring the Strength of the Relationship If the independent variable x is useful in predicting y, you will want to know how well the model fits. The strength of the relationship between x and y can be measured using:

8. Copyright ©2011 Nelson Education Limited Measuring the Strength of the Relationship Since Total SS = SSR + SSE, r 2 measures the proportion of the total variation in the responses that can be explained by using the independent variable x in the model. the percent reduction in the total variation by using the regression equation rather than just using the sample mean y-bar to estimate y.

9. Copyright ©2011 Nelson Education Limited Interpreting a Significant Regression Even if you do not reject the null hypothesis that the slope of the line equals 0, it does not necessarily mean that y and x are unrelated. Type II error—falsely declaring that the slope is 0 and that x and y are unrelated. It may happen that y and x are perfectly related in a nonlinear way.

10. Copyright ©2011 Nelson Education Limited Some Cautions You may have fit the wrong model. Extrapolation—predicting values of y outside the range of the fitted data. Causality—Do not conclude that x causes y. There may be an unknown variable at work!

11. Copyright ©2011 Nelson Education LimitedExample The table shows the heights (in cm) and weights(in Kg) of n = 10 randomly selected college football players. Player12345678910 Height, x 84.179.590.995.586.488.668.277.381.879.5 Weight, y 185.4180.3190.5182.9 190.5170.2175.3180.3175.3

12. Copyright ©2011 Nelson Education Limited Football Players