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Section 5.4

Section 5.4. Normal Distributions Finding Values. 4. 3. 2. 1. 0. 1. 2. 3. 4. From Areas to z -scores. Find the z -score corresponding to a cumulative area of 0.9803. 0.9803. 0.9803. z.

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Section 5.4

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  1. Section 5.4 Normal Distributions Finding Values

  2. 4 3 2 1 0 1 2 3 4 From Areas to z-scores Find the z-score corresponding to a cumulative area of 0.9803. 0.9803 0.9803 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. z = 2.06 corresponds roughly to the 98th percentile.

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

  4. Finding z-scores From Areas .40 .60 z z 0 Find the z-score with an area of .60 falling to its right. 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

  5. Finding z-scores From Areas -z z 0 Find the z-score such that 45% of the area under the curve falls between –z and z. .275 .275 .45 The area remaining in the tails is .55. Half this area is in each tail, so since .55/2 =.275 is the cumulative area. The closest table area is .2743 and the z-score is –0.60. Since it is a between problem, you also want to use the z-score with the opposite sign. There the z-score is -0.60 and 0.60.

  6. To find the data value, x when given a standard score, z: From z-Scores to Raw Scores The test scores for a civil service exam are normally distributed with a mean of 152 and 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

  7. Finding Percentiles or Cut-off values 10% To find the corresponding x-value, use 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 Find the cumulative area in the table that is closest to 0.1000 (the 10th percentile.) The area 0.0985 corresponds to a z-score of -1.29. *Since it is to the right we use 1.29 as our z-score.* x = 100 + 1.29(12) = 115.48. $115.48 is the smallest value for the top 10%.

  8. 1. The time spent (in days) waiting for a heart transplant in Ohio and Michigan for patients with type blood can be approximated by a normal distribution. The mean time is 127 days with a standard deviation of 23.5 days. • What is the shortest time spent waiting for a heart that would still place • a patient in the top 30% of waiting times? (b) What is the longest time spent waiting for a heart that would still place a patient in the bottom 10% of waiting times?

  9. The annual per capita use of breakfast cereal (in pounds) in the US can be • approximated by a normal distribution. The mean weight is 16.9 lbs. with a standard • deviation of 2.5 lbs. • What is the smallest annual per capita consumption of breakfast cereal that can be in the top 25% of consumptions? (b) What is the largest annual per capita consumption of breakfast cereal that can be in the bottom 15% of consumptions? Use the Standard Normal Distribution to find the z-score that corresponds to the given cumulative area or percentile. 2. 0.7454 3. 0.2912 4. P54 5. P20 Find the indicated z-score(s) indicated by the graph. 6. 7. 8. 9. Area = 0.1190 Area = 0.2266 Area = 0.0668 Area = 0.9656 on each end

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