Worked solution P29-Easy-1. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{48.5-65}{11}=-1.50,P(X\le 48.5)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-2. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{73.0-79}{6}=-1.00,P(X\le 73.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-3. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{74.0-71}{6}=0.50,P(X\le 74.0)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-4. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{77.0-62}{12}=1.25,P(X\le 77.0)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-5. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{79.0-65}{7}=2.00,P(X\le 79.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-6. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{53.0-68}{10}=-1.50,P(X\le 53.0)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-7. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{69.0-77}{8}=-1.00,P(X\le 69.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-8. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{84.5-78}{13}=0.50,P(X\le 84.5)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-9. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{72.2-61}{9}=1.25,P(X\le 72.2)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-10. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{100.0-80}{10}=2.00,P(X\le 100.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-11. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{64.5-75}{7}=-1.50,P(X\le 64.5)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-12. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{56.0-64}{8}=-1.00,P(X\le 56.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-13. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{66.5-60}{13}=0.50,P(X\le 66.5)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-14. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{76.8-68}{7}=1.25,P(X\le 76.8)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-15. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{83.0-63}{10}=2.00,P(X\le 83.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-16. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{54.5-71}{11}=-1.50,P(X\le 54.5)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-17. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{72.0-78}{6}=-1.00,P(X\le 72.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-18. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{84.0-78}{12}=0.50,P(X\le 84.0)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-19. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{75.0-65}{8}=1.25,P(X\le 75.0)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Easy-20. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{93.0-75}{9}=2.00,P(X\le 93.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-1. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{49.5-66}{11}=-1.50,P(X\le 49.5)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-2. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{49.0-61}{12}=-1.00,P(X\le 49.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-3. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{69.0-65}{8}=0.50,P(X\le 69.0)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-4. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{73.8-65}{7}=1.25,P(X\le 73.8)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-5. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{87.0-69}{9}=2.00,P(X\le 87.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-6. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{58.5-75}{11}=-1.50,P(X\le 58.5)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-7. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{66.0-78}{12}=-1.00,P(X\le 66.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-8. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{65.5-61}{9}=0.50,P(X\le 65.5)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-9. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{71.5-64}{6}=1.25,P(X\le 71.5)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-10. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{100.0-78}{11}=2.00,P(X\le 100.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-11. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{60.0-78}{12}=-1.50,P(X\le 60.0)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-12. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{56.0-69}{13}=-1.00,P(X\le 56.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-13. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{66.0-62}{8}=0.50,P(X\le 66.0)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-14. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{76.5-69}{6}=1.25,P(X\le 76.5)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-15. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{99.0-75}{12}=2.00,P(X\le 99.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-16. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{60.5-80}{13}=-1.50,P(X\le 60.5)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-17. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{67.0-75}{8}=-1.00,P(X\le 67.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-18. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{75.0-72}{6}=0.50,P(X\le 75.0)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-19. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{74.5-67}{6}=1.25,P(X\le 74.5)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Tough-20. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{95.0-77}{9}=2.00,P(X\le 95.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-1. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{48.0-63}{10}=-1.50,P(X\le 48.0)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-2. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{59.0-68}{9}=-1.00,P(X\le 59.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-3. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{83.0-79}{8}=0.50,P(X\le 83.0)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-4. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{88.0-73}{12}=1.25,P(X\le 88.0)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-5. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{82.0-62}{10}=2.00,P(X\le 82.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-6. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{70.0-79}{6}=-1.50,P(X\le 70.0)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-7. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{61.0-74}{13}=-1.00,P(X\le 61.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-8. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{76.5-70}{13}=0.50,P(X\le 76.5)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-9. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{79.0-64}{12}=1.25,P(X\le 79.0)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-10. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{87.0-71}{8}=2.00,P(X\le 87.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-11. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{49.0-64}{10}=-1.50,P(X\le 49.0)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-12. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{62.0-70}{8}=-1.00,P(X\le 62.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-13. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{72.5-68}{9}=0.50,P(X\le 72.5)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-14. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{77.8-64}{11}=1.25,P(X\le 77.8)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-15. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{100.0-76}{12}=2.00,P(X\le 100.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-16. The standardized score is z=-1.50; the modeled proportion at or below x is approximately 0.0668. $$z=\frac{51.0-66}{10}=-1.50,P(X\le 51.0)=\Phi (-1.50)\approx 0.0668.$$ Interpretation: The value is 1.50 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-17. The standardized score is z=-1.00; the modeled proportion at or below x is approximately 0.1587. $$z=\frac{53.0-65}{12}=-1.00,P(X\le 53.0)=\Phi (-1.00)\approx 0.1587.$$ Interpretation: The value is 1.00 standard deviations below the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-18. The standardized score is z=0.50; the modeled proportion at or below x is approximately 0.6915. $$z=\frac{74.5-70}{9}=0.50,P(X\le 74.5)=\Phi (0.50)\approx 0.6915.$$ Interpretation: The value is 0.50 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-19. The standardized score is z=1.25; the modeled proportion at or below x is approximately 0.8944. $$z=\frac{79.8-71}{7}=1.25,P(X\le 79.8)=\Phi (1.25)\approx 0.8944.$$ Interpretation: The value is 1.25 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.

Worked solution P29-Toughest-20. The standardized score is z=2.00; the modeled proportion at or below x is approximately 0.9772. $$z=\frac{79.0-63}{8}=2.00,P(X\le 79.0)=\Phi (2.00)\approx 0.9772.$$ Interpretation: The value is 2.00 standard deviations above the model mean. Validity: A normal probability statement requires a reasonable normal model; standardization alone does not establish shape. Error to reject: A z-score is a signed standard-deviation distance, whereas a percentile is a cumulative proportion below a value.