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.
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Find the z-score corresponding to a cumulative area of 0.9803.
z = 2.06 corresponds
roughly to the
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.
Find the z-score corresponding to the 90th percentile.
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.
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
Finding z-Scores from Areas
Find the z-score such that 45% of the area under the curve falls between –z and 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.
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
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%.
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.