1. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Chapter Correlation and Regression 9

2. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 2 Chapter Outline 9.1 Correlation 9.2 Linear Regression 9.3 Measures of Regression and Prediction Intervals 9.4 Multiple Regression.

3. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 3 Section 9.3 Measures of Regression and Prediction Intervals.

4. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 4 Section 9.3 Objectives How to interpret the three types of variation about a regression line How to find and interpret the coefficient of determination How to find and interpret the standard error of the estimate for a regression line How to construct and interpret a prediction interval for y.

5. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 5 Variation About a Regression Line Three types of variation about a regression line  Total variation  Explained variation  Unexplained variation To find the total variation, you must first calculate  The total deviation  The explained deviation  The unexplained deviation.

6. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 6 Variation About a Regression Line (x i, ŷ i ) x y (x i, y i ) Unexplained deviation Total deviation Explained deviation Total Deviation = Explained Deviation = Unexplained Deviation =.

7. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 7 Total variation The sum of the squares of the differences between the y-value of each ordered pair and the mean of y. Explained variation The sum of the squares of the differences between each predicted y-value and the mean of y. Variation About a Regression Line Total variation = Explained variation =.

8. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 8 Unexplained variation The sum of the squares of the differences between the y-value of each ordered pair and each corresponding predicted y-value. Variation About a Regression Line Unexplained variation = The sum of the explained and unexplained variation is equal to the total variation. Total variation = Explained variation + Unexplained variation.

9. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 9 Coefficient of Determination Coefficient of determination The ratio of the explained variation to the total variation. Denoted by r 2.

10. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 10 Example: Coefficient of Determination About 78% of the variation in the carbon emissions can be explained by the variation in the gross domestic products. About 22% of the variation is unexplained. The correlation coefficient for the gross domestic products and carbon dioxide emissions data is r ≈ 0.883. Find the coefficient of determination. What does this tell you about the explained variation of the data about the regression line? About the unexplained variation? Solution:.

11. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 11 The Standard Error of Estimate Standard error of estimate The standard deviation of the observed y i -values about the predicted ŷ-value for a given x i -value. Denoted by s e. The closer the observed y-values are to the predicted y-values, the smaller the standard error of estimate will be. n is the number of ordered pairs in the data set.

12. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 12 The Standard Error of Estimate 1.Make a table that includes the column heading shown. 2.Use the regression equation to calculate the predicted y-values. 3.Calculate the sum of the squares of the differences between each observed y-value and the corresponding predicted y-value. 4.Find the standard error of estimate. In WordsIn Symbols.

13. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 13 Example: Standard Error of Estimate The regression equation for the gross domestic products and carbon dioxide emissions data as calculated in section 9.2 is ŷ = 196.152x + 102.289 Find the standard error of estimate. Solution: Use a table to calculate the sum of the squared differences of each observed y-value and the corresponding predicted y-value..

14. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 14 Solution: Standard Error of Estimate xyŷ iŷ i (y i – ŷ i ) 2 1.6428.2416.1322145.63179684 3.6828.8808.4362414.68435044 4.91214.21063.433822,730.44706244 1.1444.6318.056216,013.33331844 0.9264.0278.8258219.80434564 2.9415.3671.129865,448.88656804 2.7571.8631.89943611.93788036 2.3454.9553.43869709.85568996 1.6358.7416.13223298.45759684 1.5573.5396.51731,322.982289 Σ = 152,916.020898 unexplained variation.

15. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 15 Solution: Standard Error of Estimate n = 10, Σ(y i – ŷ i ) 2 = 152,916.020898 The standard error of estimate of the carbon dioxide emissions for a specific gross domestic product is about 138.255 million metric tons..

16. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 16 Prediction Intervals Two variables have a bivariate normal distribution if for any fixed value of x, the corresponding values of y are normally distributed and for any fixed values of y, the corresponding x-values are normally distributed..

17. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 17 Prediction Intervals A prediction interval can be constructed for the true value of y. Given a linear regression equation ŷ = mx + b and x 0, a specific value of x, a c-prediction interval for y is ŷ – E < y < ŷ + E where The point estimate is ŷ and the margin of error is E. The probability that the prediction interval contains y is c..

18. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 18 Constructing a Prediction Interval for y for a Specific Value of x 1.Identify the number of ordered pairs in the data set n and the degrees of freedom. 2.Use the regression equation and the given x-value to find the point estimate ŷ. 3.Find the critical value t c that corresponds to the given level of confidence c. Use Table 5 in Appendix B. In WordsIn Symbols d.f. = n – 2.

19. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 19 Constructing a Prediction Interval for y for a Specific Value of x 4.Find the standard error of estimate s e. 5.Find the margin of error E. 6.Find the left and right endpoints and form the prediction interval. In WordsIn Symbols Left endpoint: ŷ – E Right endpoint: ŷ + E Interval: ŷ – E < y < ŷ + E.

20. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 20 Example: Constructing a Prediction Interval Construct a 95% prediction interval for the carbon dioxide emission when the gross domestic product is $3.5 trillion. What can you conclude? Recall, n = 10, ŷ = 196.152x + 102.289, s e = 138.255 Solution: Point estimate: ŷ = 196.152(3.5) + 102.289 ≈ 788.821 Critical value: d.f. = n –2 = 10 – 2 = 8 t c = 2.306.

21. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 21 Solution: Constructing a Prediction Interval Left Endpoint: ŷ – ERight Endpoint: ŷ + E 439.397 < y < 1138.245 788.821 – 349.424 ≈ 439.397 788.821 + 349.424 ≈ 1138.245 You can be 95% confident that when the gross domestic product is $3.5 trillion, the carbon dioxide emissions will be between 439.397 and 1138.245 million metric tons..

22. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 22 Section 9.3 Summary Interpreted the three types of variation about a regression line Found and interpreted the coefficient of determination Found and interpreted the standard error of the estimate for a regression line Constructed and interpreted a prediction interval for y.