Download Notes 6A – Characterizing Data

Notes 6C – The Normal Distribution
Normal Distribution – a symmetric, bell-shaped distribution with a single peak. Its peak corresponds to the
mean, median, and mode of the distribution.
Example 1 The Normal Shape
Figure 6.14a shows a famous data set of the chest sizes of 5738 Scottish militiamen collected in about 1846.
Figure 6.14b shows the distribution of the population densities of the 50 states. Which distribution appears to
be normal? Explain.
Now try Exercises 11-12
Example 2 Is It a Normal Distribution?
Which of the following variables would you expect to have a normal or nearly normal distribution?
a) Scores on a very easy test
b) Shoe sizes of a random sample of adult women
Now try Exercises 11-12
Example 3 SAT Scores
Each test that makes up the SAT is designed so that its scores are normally distributed with a mean of 500 and a
standard deviation of 100. Interpret this statement according to the 68-95-99.7 rule.
Now try Exercise 19
Example 4 Detecting Counterfeits
Vending machines can be adjusted to reject coins above and below certain weights. The weights of legal U.S.
quarters are normally distributed with a mean of 5.67 grams and a standard deviation of 0.0700 gram. If a
vending machine is adjusted to reject quarters that weigh more than 5.81 grams and less than 5.53 grams, what
percentage of legal quarters will be rejected by the machine.
Now try Exercise 20
Example 5 Normal Auto Prices
A survey finds that the prices paid for two-year-old Ford Fusion cars are normally distributed with a mean of
$10,500 and a standard deviation of $500. Considers a sample of 10,000 people who bought two-year-old Ford
Fusions.
a) How many people paid between $10,000 and $11,000?
b) How many paid less than $10,000?
c) How many paid more than $12,000?
Now try Exercises 21-24
Example 6 Standard IQ Scores
The Stanford-Binet IQ test is designed so that scores are normally distributed with a mean of 100 and a standard
deviation of 16. Find the standard scores for IQ scores of 95 and 125.
Now try Exercises 25-28
Percentiles – The nth percentile of a data set is the smallest value in the set with the property that n% of the
data values are less than or equal to it. A data vale that lies between two percentiles is said to lie in the lower
percentile.
Table 6.3
Example 7 Cholesterol Levels
Cholesterol levels in men 18 to 24 years of age are normally distributed with a mean of 178 and a standard
deviation of 41.
a) In what percentile is a man with a cholesterol level of 190?
b) What cholesterol level corresponds to the 90th percentile, the level at which treatment may be necessary?
Now try Exercises 29-30
Example 8 Women in the Army
The heights for American women aged 18 to 24 are normally distributed with a mean of 65 inches and a
standard deviation of 2.5 inches. In order to serve in the U.S. Army, women must be between 58 inches and 80
inches tall. What percentage of women are ineligible to serve based on their heights?
Now try Exercises 31-32