From Areas to z -Scores

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# From Areas to z -Scores - PowerPoint PPT Presentation

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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Presentation Transcript
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.

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.

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

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.

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

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.