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Sec 10.3

Sec 10.3. Coefficient of Determination and Standard Error of the Estimate. Review concepts. fill in the third row of the table for each x value. Review concepts. Test 10-11 Friday April 18. Monday April 7 Periods 5-8 assembly schedule. Monday 21 periods 5-8 Wed schedule.

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Sec 10.3

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  1. Sec 10.3 Coefficient of Determination and Standard Error of the Estimate.

  2. Review concepts fill in the third row of the table for each x value.

  3. Review concepts

  4. Test 10-11 Friday April 18 Monday April 7 Periods 5-8 assembly schedule Monday 21 periods 5-8 Wed schedule. Wed April 23 ACT Our class doesn’t meet

  5. Variation Regression line Bluman, Chapter 10

  6. Variations • The total variation is calculated by: This the sum of the squares of the vertical distances from the mean.

  7. 2 parts of variation • The total variation is made up off two types of variation: • Explained variation: attributed to the relationship between x & y. • Unexplained variation: due to chance.

  8. Explained variation Most of the variation can be explained by the relationship.

  9. Unexplained variation When this variation is small then the value of r will be close to 1 or -1.

  10. The last few slides summarized: • Total variation Unexplained variation Explained variation

  11. Residuals • The values of (y-y') are called residuals. • A residual is the difference between the actual value of y and the predicted value of y' for a given x value. • The mean of the residuals is always zero.

  12. Coefficient of . . . • Determination, r2 • non-determination, 1-r2

  13. Coefficient of determination, r2 • The coefficient of determination,r2, is a measure of the variation of the dependent variable that is explained by the regression line and the independent variable.

  14. Coefficient of Determiation • The coefficient of determination is the ratio of the explained variation to the total variation. • The symbol for the coefficient of determination is r 2. • Another way to arrive at the value for r 2is to square the correlation coefficient. Bluman, Chapter 10

  15. Coefficient of Nondetermination: • Coefficient nondetermination is the measure of the rest of the variation that is not explained by r2. • It is the complement of r2 and equals to 1-r2.

  16. Coefficient of Nondetermiation • The coefficient of nondeterminationis a measure of the unexplained variation. • The formula for the coefficient of determination is 1.00 – r 2. Bluman, Chapter 10

  17. Some facts: • The coefficient of determination is a percent. • i.e. if r2=.81 that means 81% of variation in the dependent variable is explained by the variation in the independent variable. • The coefficient of nondetermination is: • 1-81%=19% and it means that 19% …

  18. Example: • Let r=0.9123 • Find the coefficients of determination and nondetermination. • Explain the meaning of each.

  19. Standard Error of the estimate • Symbol: Sest • Sest is the standard deviation of the observed y values about the predicted y' values.

  20. Standard Error of the Estimate • The standard error of estimate, denoted by sest is the standard deviation of the observed y values about the predicted y' values. The formula for the standard error of estimate is: Bluman, Chapter 10

  21. Chapter 10Correlation and Regression Section 10-3 Example 10-12 Page #569 Bluman, Chapter 10

  22. Example 10-12: Copy Machine Costs A researcher collects the following data and determines that there is a significant relationship between the age of a copy machine and its monthly maintenance cost. The regression equation is y  = 55.57 + 8.13x. Find the standard error of the estimate. Bluman, Chapter 10

  23. Example 10-12: Copy Machine Costs x Age Monthly y y y – y (y – y 2 Machine (years) cost, ¢ ¢ ¢ ) A 1 62 63.70 - 1.70 2.89 B 2 78 71.83 6.17 38.0689 C 3 70 79.96 - 9.96 99.2016 D 4 90 88.09 1.91 3.6481 E 4 93 88.09 4.91 24.1081 F 6 103 104.35 - 1.35 1.8225 169.7392 Bluman, Chapter 10

  24. Chapter 10Correlation and Regression Section 10-3 Example 10-13 Page #570 Bluman, Chapter 10

  25. Example 10-13: Copy Machine Costs Bluman, Chapter 10

  26. Example 10-13: Copy Machine Costs x Age Monthly y xy y2 Machine (years) cost, A 1 62 62 3,844 B 2 78 156 6,084 C 3 70 210 4,900 D 4 90 360 8,100 E 4 93 372 8,649 F 6 103 618 10,609 1778 496 42,186 Bluman, Chapter 10

  27. Formula for the Prediction Interval about a Value y  Bluman, Chapter 10

  28. Chapter 10Correlation and Regression Section 10-3 Example 10-14 Page #571 Bluman, Chapter 10

  29. Example 10-14: Copy Machine Costs For the data in Example 10–12, find the 95% prediction interval for the monthly maintenance cost of a machine that is 3 years old. Step 1: Find Step 2: Find y for x= 3. Step 3: Find sest. (as shown in Example 10-13) Bluman, Chapter 10

  30. Example 10-14: Copy Machine Costs Step 4: Substitute in the formula and solve. Bluman, Chapter 10

  31. Example 10-14: Copy Machine Costs Step 4: Substitute in the formula and solve. Hence, you can be 95% confident that the interval 60.53 < y < 99.39 containsthe actual value of y. Bluman, Chapter 10

  32. Read section 10.3 • Take notes on Residuals. • Review the calculator steps. • Page 574 • #1-7 all, 9-17 odds Bluman, Chapter 10

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