Section 5.4

1. Section 5.4 Normal Distributions Finding Values

2. From Areas to z-Scores Find the z-score corresponding to a cumulative area of 0.9803. 0.9803 –4 –3 –2 –1 0 1 2 3 4 z

3. From Areas to z-Scores Find the z-score corresponding to a cumulative area of 0.9803. z = 2.06 corresponds roughly to the 98th percentile. 0.9803 –4 –3 –2 –1 0 1 2 3 4 z Locate 0.9803 in the area portion of the table. Read the values at the beginning of the corresponding row and at the top of the column. The z-score is 2.06.

4. Class Practice p234 1-11 odd

5. -2.05 0.85 5. -0.16 2.39 -1.645 11. 0.84

6. Finding z-Scores from Areas Find the z-score corresponding to the 90th percentile. .90 z 0

7. Finding z-Scores from Areas Find the z-score corresponding to the 90th percentile. .90 z 0 The closest table area is .8997. The row heading is 1.2 and column heading is .08. This corresponds to z = 1.28. A z-score of 1.28 corresponds to the 90th percentile.

8. Class Practice P234 13-23 odd

9. 13. -2.325 -0.25 17. 1.175 19. -0.675 21. 0.675 23. -0.385

10. Finding z-Scores from Areas Find the z-score with an area of .60 falling to its right. .60 z 0 z

11. Finding z-Scores from Areas Find the z-score with an area of .60 falling to its right. .40 .60 z 0 z With .60 to the right, cumulative area is .40. The closest area is .4013. The row heading is 0.2 and column heading is .05. The z-score is 0.25. A z-score of 0.25 has an area of .60 to its right. It also corresponds to the 40th percentile

12. 0 Finding z-Scores from Areas Find the z-score such that 45% of the area under the curve falls between –z and z. .45 –z z

13. 0 Finding z-Scores from Areas Find the z-score such that 45% of the area under the curve falls between –z and z. .275 .275 .45 –z z The area remaining in the tails is .55. Half this area is in each tail, so since .55/2 = .275 is the cumulative area for the negative z value and .275 + .45 = .725 is the cumulative area for the positive z. The closest table area is .2743 and the z-score is 0.60. The positive z score is 0.60.

14. Class Practice P 234 27-33 odd

15. 27. -1.645, 1.645 29. -1.96, 1.96 31. 0.325 33. 1.28

16. From z-Scores to Raw Scores To find the data value, x when given a standard score, z: The test scores for a civil service exam are normally distributed with a mean of 152 and a standard deviation of 7. Find the test score for a person with a standard score of: (a) 2.33 (b) –1.75 (c) 0

17. From z-Scores to Raw Scores To find the data value, x when given a standard score, z: The test scores for a civil service exam are normally distributed with a mean of 152 and a standard deviation of 7. Find the test score for a person with a standard score of: (a) 2.33 (b) –1.75 (c) 0 (a) x = 152 + (2.33)(7) = 168.31 (b) x = 152 + (–1.75)(7) = 139.75 (c) x = 152 + (0)(7) = 152

18. Finding Percentiles or Cut-off Values Monthly utility bills in a certain city are normally distributed with a mean of $100 and a standard deviation of $12. What is the smallest utility bill that can be in the top 10% of the bills? z

19. Finding Percentiles or Cut-off Values Monthly utility bills in a certain city are normally distributed with a mean of $100 and a standard deviation of $12. What is the smallest utility bill that can be in the top 10% of the bills? $115.36 is the smallest value for the top 10%. 90% 10% z Find the cumulative area in the table that is closest to 0.9000 (the 90th percentile.) The area 0.8997 corresponds to a z-score of 1.28. To find the corresponding x-value, use x = 100 + 1.28(12) = 115.36.

20. Class Practice p 235 35 and 39

21. 35. a. 68.52 b. 62.14 39. a. 139.22 b. 96.92 HW p 234-236 2-40 even