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Section 9.3 Transformations of Data

Section 9.3 Transformations of Data. Objectives: 1. To find z-scores and other transformed scores for data. 2. To determine the effect of transformations on the mean and standard deviation. Original . Transformed. 4 6 7 9 9 10. 9 11 12 14 14 15. Mean 7.5

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Section 9.3 Transformations of Data

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  1. Section 9.3 Transformations of Data

  2. Objectives: 1. To find z-scores and other transformed scores for data. 2. To determine the effect of transformations on the mean and standard deviation.

  3. Original Transformed 4 6 7 9 9 10 9 11 12 14 14 15 Mean7.5 St Dev2.26 12.5 2.26

  4. Theorem 9.1: Translated Data Let x1, x2, . . ., xnrepresent the original data. If y1, y2, . . ., ynare obtained by adding a constant k to the original data values, then y = x + k and sy = sx.

  5. Original Transformed 4 6 7 9 9 10 12 18 21 27 27 30 Mean7.5 St Dev2.26 22.5 6.77

  6. If y1, y2, . . ., ynare found from x1, x2, . . ., xn by multiplying each by the same constant k, then y =kx and sy = ksx. Theorem 9.2: Scaled Data

  7. x - x z = s Definition z-score The transformed score found by subtracting the mean from the individual score, and dividing by the standard deviation:

  8. The statistic z is a measure of the deviation of an individual score from the mean in units of standard deviation.

  9. x - x 135 - 100 = z = 16 s Practice: If you scored a 135 on an IQ test that has a mean of 100 and standard deviation of 16, how many standard deviations are you away from the mean? ≈2.2

  10. A z-score of 2.2 on the IQ test means that the score earned (135) was 2.2 standard deviations (16) above the mean (100). If the individual score x is the same as the mean, the z-score is 0. A score below the mean will result in a negative z-score.

  11. Practice: Use the given set of scores to find the following values. 63, 74, 67, 76, 77, 71, 68, 66 Find the mean.

  12. Practice: Use the given set of scores to find the following values. 63, 74, 67, 76, 77, 71, 68, 66 Find the standard deviation.

  13. Practice: Use the given set of scores to find the following values. 63, 74, 67, 76, 77, 71, 68, 66 Find the z-score of the lowest score.

  14. Practice: Use the given set of scores to find the following values. 63, 74, 67, 76, 77, 71, 68, 66 Find the z-score of the highest score.

  15. Practice: Use the given set of scores to find the following values. 63, 74, 67, 76, 77, 71, 68, 66 Transform the rest of the scores to z-scores.

  16. Homework: pp. 460-461

  17. If the mean is 83 and the standard deviation is 7, find the z-score for each test score below. 1. 88

  18. If the mean is 83 and the standard deviation is 7, find the z-score for each test score below. 5. 73

  19. If the mean of a set of data values is 75 and the standard deviation is 10, find the mean and standard deviation for the data transformed as follows. 9. y = 1/5x - 10

  20. SAT scores are calculated from z-scores using the transformation SAT = 100z+500. 16. Give the mean score on the SAT.

  21. SAT scores are calculated from z-scores using the transformation SAT = 100z+500. 17. Give the standard deviation on the SAT.

  22. SAT scores are calculated from z-scores using the transformation SAT = 100z+500. 18. Give the SAT score of someone who scored 2.5 standard deviations above the mean.

  23. SAT scores are calculated from z-scores using the transformation SAT = 100z+500. 19. What does an SAT score of 563 mean?

  24. ■ Cumulative Review 26. A line passes through (3, 4) with an angle of inclination of 20°. Write its equation in slope- intercept form.

  25. ■ Cumulative Review 27. State the three Pythagorean identities.

  26. ■ Cumulative Review 28. In class, three quiz scores range from 10 to 20 with a median of 18. Find the mean, midrange, and mode.

  27. ■ Cumulative Review 29. Graph r = 2 + 3 cos .

  28. ■ Cumulative Review 30. If sin  = , give the other five trig functions in terms of a and b. a b

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