Create Presentation
Download Presentation

Download Presentation

The Coefficient of Determination

Download Presentation
## The Coefficient of Determination

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**The Coefficient of Determination**The coefficient of determination, r2,is the ratio of explained variation in y to the total variation in y. The correlation coefficient of number of times absent and final grade is r = –0.975. The coefficient of determination is r2 = (–0.975)2 = 0.9506. Interpretation: About 95% of the variation in final grades can be explained by the number of times a student is absent. The other 5% is unexplained and can be due to sampling error or other variables such as intelligence, amount of time studied, etc.**The Standard Error of Estimate**The Standard Error of Estimate,se,is the standard deviation of the observed yi values about the predicted value.**The Standard Error of Estimate**x y 1 8 78 74.275 13.8756 2 2 92 97.819 33.8608 3 5 90 86.047 15.6262 4 12 58 58.579 0.3352 5 15 43 46.807 14.4932 6 9 74 70.351 13.3152 7 6 81 82.123 1.2611 92.767 Calculate for each x. = 4.307**Prediction Intervals**Given a specific linear regression equation and x0, a specific value of x, a c-prediction interval for y is: where The point estimate is andEis the maximum error of estimate. Use a t-distribution with n – 2 degrees of freedom.**Application**Construct a 90% confidence interval for a final grade when a student has been absent 6 times. 1. Find the point estimate: The point (6, 82.123) is the point on the regression line with x-coordinate of 6.**Application**Construct a 90% confidence interval for a final grade when a student has been absent 6 times. 2. Find E, At the 90% level of confidence, the maximum error of estimate is 9.438.**Application**Construct a 90% confidence interval for a final grade when a student has been absent 6 times. 3. Find the endpoints. – E = 82.123 – 9.438 = 72.685 + E = 82.123 + 9.438 = 91.561 72.685 < y < 91.561 When x = 6, the 90% confidence interval is from 72.685 to 91.586.**Minitab Output**Regression Analysis The regression equation is y = 106 – 3.92x Predictor Coef StDev T P Constant 105.668 3.655 28.91 0.000 x –3.9241 0.4019 –9.76 0.000 S = 4.307 R-Sq = 95.0% R-Sq(adj) = 94.0%