Section 5.4 Normal Distributions Finding Values

From Areas to z-Scores z –1 1 2 3 4 Find the z-score corresponding to a cumulative area of 0.9803. 0.9803 Be sure to emphasize that here, the area is given. Tell students to choose the z score closest to the given area. The only exception is if the area falls exactly at the midpoint between two z-scores, use the midpoint of the z=scores. –4 –3 –2 –1 1 2 3 4 z

From Areas to z-Scores z –1 1 2 3 4 Find the z-score corresponding to a cumulative area of 0.9803. z = 2.06 corresponds roughly to the 98th percentile. 0.9803 Be sure to emphasize that here, the area is given. Tell students to choose the z score closest to the given area. The only exception is if the area falls exactly at the midpoint between two z-scores, use the midpoint of the z=scores. –4 –3 –2 –1 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.

Class Practice p234 1-11 odd

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

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

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

Class Practice P234 13-23 odd

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

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

Finding z-Scores from Areas Find the z-score with an area of .60 falling to its right. .40 .60 z 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

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 Because the normal distribution is symmetric, the z scores will have the same absolute value. As a result, you can find one z-score and use its opposite for the other.

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. Because the normal distribution is symmetric, the z scores will have the same absolute value. As a result, you can find one z-score and use its opposite for the other.

Class Practice P 234 27-33 odd

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

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 Show students that the formula given is equivalent to the z-score formula. Some students prefer to use only one formula and others like to use both. Have students work these through before displaying the answers. Emphasize the meaning of z-scores. A z-score of 2.33 is a 2.33 standard deviations above the mean.

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 Show students that the formula given is equivalent to the z-score formula. Some students prefer to use only one formula and others like to use both. Have students work these through before displaying the answers. Emphasize the meaning of z-scores. A z-score of 2.33 is a 2.33 standard deviations above the mean. (b) x = 152 + (–1.75)(7) = 139.75 (c) x = 152 + (0)(7) = 152

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 Students find these “cut-off” problems easier if they think in terms of percentiles, which in turn are interpreted as cumulative areas.

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 Students find these “cut-off” problems easier if they think in terms of percentiles, which in turn are interpreted as cumulative areas. 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.

Class Practice p 235 35 and 39

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