Chapter 5: z-Scores

Chapter 5: z-Scores 1. A z-score describes a precise location within a distribution. The sign of the z-score tells whether the location is above (+) or below (–) the mean, and the magnitude tells the distance from the mean in terms of the number of standard deviations. 2. 3. a. b. c. d. z = 0.50 z = 2.00 z = –1.00 z = –1.50 a. b. c. d. above the mean by 8 points above the mean by 2 points below the mean by 8 points below the mean by 2 points 4. a. b. 5. X z 55 0.50 45 –0.50 X z 60 1.00 30 –2.00 X z 60 1.00 45 –0.50 X z 58 0.80 47 –0.30 X z 65 1.50 35 –1.50 X 70 75 X z 53 –0.78 58 –0.22 X 65 48 6. a. b. 7. a. z 0.56 –1.33 z 1.11 1.67 X z 75 2.50 35 –1.50 b. X z 49 0.75 37 –0.25 X z 58 1.50 34 –0.50 X z 16 –2.00 55 1.25 8. X z 44 1.00 28 –3.00 X z 42 0.50 50 2.50 X z 48 2.00 46 1.50 9. X 84 78 10. a. b. c. d. z = +1.00 z = +2.00 z = –0.50 z = –2.00 11. a. b. c. d. X = 49 X = 70 X = 86 X = 92 X z 86 1.20 77 –0.60 X z 90 2.00 71 –1.80 12. σ = 3 13. σ = 8 14. μ = 49 X z 105 0.50 90 –1.00 X z 120 2.00 85 –1.50 X z 130 3.00 60 –4.00 15. M = 56 X z 90 –1.00 107 0.70 X z 95 –0.50 115 1.50 X z 120 2.00 85 –1.50 17. σ = 3 X z 49 0.75 28 –1.00 X z 52 1.00 64 2.00 X z 43 0.25 34 –0.50 z 0.80 –0.40 16. s = 10 18. μ = 82 and σ = 3. The distance between the two scores is 9 points which is equal to 3 standard deviations. 19. μ = 60 and σ = 4. The distance between the two scores is 10 points which is equal to 2.5 standard deviations. 20. a. central (z = 0.50) b. extreme (z = 5.00) c. extreme (z = –2.50) d. central (z = –0.50) 21. σ = 2 22. σ = 10 23. X = 43 corresponds to z = 1.50, and X = 60 corresponds to z = 0.50. X = 43 has a higher position in its distribution and should receive the higher grade. 27. a. μ = 4 and σ = 2 b. & c. Original X 6 1 3 4 7 3 z-score Transformed X 1.00 60 –1.50 35 –0.50 45 0 50 1.50 65 –0.50 45 28. a. μ = 5 and σ = 4 b. & c. Original X 0 6 4 3 12 z-score Transformed X –1.25 50 0.25 62 –0.25 58 –0.50 56 1.75 74 24. X = 55 corresponds to z = –1.00, and X = 40 corresponds to z = –0.50. X = 40 has a higher position in its distribution and should receive the higher grade. 25. a. b. c. d. X = 80 (z = –1.00) X = 40 (z = –3.00) X = 140 (z = 2.00) X = 110 (z = 0.50) 26. a. b. c. d. X = 55 X = 51 X = 46 X = 40 (z = 0.50) (z = 0.10) (z = –0.40) (z = –1.00 29. a. X z ──────── 10 +1.00 0 –1.50 4 –0.50 8 +0.50 4 –0.50 10 +1.00 b. For the sample of n = 6 z-scores, Σz = 0 and M = 0. Also for the z-scores, SS = 5, s2 = 5/5 = 1, and s = 1.