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### Section 5.3

Normal Distributions: Finding Values

Examples 1 & 2

- Find the z-score that corresponds to a cumulative area of 0.3632.
- Find the z-score that has 10.75% of the distribution’s area to its right.

Example 3 & 4

- Find the z-score that has 96.16% of the distribution’s area to the right.
- Find the z-score for which 95% of the distribution’s area lies between z and –z.

Examples 5 – 7

- P5
- P50
- P90

Transforming a z-Score to an x-Value

- Definition 1: Transforming a z-Score to an x-Value:
- To transform a standard z-score to a data value x in a given population, use the formula

Example 8

The speeds of vehicles along a stretch of highway are normally distributed, with a mean of 56 miles per hour and a standard deviation of 4 miles per hour. Find the speeds x corresponding to z-scores of 1.96, -2.33, and 0. Interpret your results.

- 63.84 is above the mean,
- 46.68 is below the mean,
- 56 is the mean.

TOTD

- Use the Standard Normal Table to find the z-score that corresponds to the given cumulative area or percentile.
- Find the indicated z-score.
- Find the z-score that has 78.5% of the distribution’s area to its right.

Example 9

The monthly utility bills in a city are normally distributed, with a mean of $70 and a standard deviation of $8. Find the x-values that correspond to z-scores of -0.75, 4.29, and -1.82. What can you conclude?

- Negative z-scores represent
bills that are lower than the

mean.

Example 10

Scores for a civil service exam are normally distributed, with a mean of 75 and a standard deviation of 6.5. To be eligible for civil service employment, you must score in the top 5%. What is the lowest score you can earn and still be eligible for employment?

- The lowest score you can earn and still
be eligible for employment is 86.

Example 11

The braking distances of a sample of Ford F-150s are normally distributed. On a dry surface, the mean braking distance was 158 feet and the standard deviation was 6.51 feet. What is the longest braking distance on a dry surface one of these Ford F-150s could have and still be in the top 1%?

- The longest breaking
distance on a dry surface

for an F-150 in the top 1%

is 143 ft.

Example 12

In a randomly selected sample of 1169 men ages 35-44, the mean total cholesterol level was 205 milligrams per deciliter with a standard deviation of 39.2 milligrams per deciliter. Assume the total cholesterol levels are normally distributed. Find the highest total cholesterol level a man in this 35-44 age group can have and be in the lowest 1%.

- The value that separates the lowest 1%
of total cholesterol levels for men in the

35 – 44 age group from the highest 99%

is about 114.

Example 13

The length of time employees have worked at a corporation is normally distributed, with a mean of 11.2 years and a standard deviation of 2.1 years. In a company cutback, the lowest 10% in seniority are laid off. What is the maximum length of time an employee could have worked and still be laid off?

- The maximum length of time an
employee could have worked and

still be laid off is 8.5 years.

TOTD

- Find the indicated area under the standard normal curve.
- To the right of z = 1.645
- Between z = -1.53 and z = 0

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