Statistics
5.3 Normal Distributions: Finding Values
LEQ: How do you find a specific data value for a given probability?
Procedure:
1. Finding z-Scores:
a. Examples 1 – 4: Finding a z-Score given an area.
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.
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.
b. Examples 5 – 7: Finding a z-score given a percentile.
Find the z-score that corresponds to each percentile.
5. P5
6. P50
7. P90
2. Transforming a z-Score to an x-Value:
c. 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
d. Example 8 & 9: Finding an x-value corresponding to a z-score.
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.
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?
3. Finding a Specific Data Value for a Given Probability:
e. Example 10 & 11: Finding a specific data value.
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?
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%?
f. Examples 12 & 13: Finding a specific data value.
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%.
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?
2. HW: p. 242 (3 – 45 mo3)