#### Transcript Slide 1

```Section 5.3
Normal Distributions: Finding
Values
Examples 1 & 2
1.
Find the z-score that corresponds to a
cumulative area of 0.3632.
2.
Find the z-score that has 10.75% of the
distribution’s area to its right.
Example 3 & 4
3.
Find the z-score that has 96.16% of the
distribution’s area to the right.
4.
Find the z-score for which 95% of the distribution’s
area lies between z and –z.
Examples 5 – 7
5.
P5
6.
P50
7.
P90
Transforming a z-Score to an xValue
 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
𝑥 = 𝜇 + 𝑧𝜎
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.
𝑃35

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%
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
```