Download normal distribution

Putting Statistics to Work
Discussion Paragraph 6B
1 web
26. Web Data Sets
1 world
27. Ranges in the News
28. Summarizing a News Data
Set
29. Range Rule in the News
Copyright © 2011 Pearson Education, Inc.
Unit 6C
The Normal Distribution
Copyright © 2011 Pearson Education, Inc.
Slide 6-3
6-C
The Normal Distribution
The normal distribution is a symmetric, bellshaped distribution with a single peak. Its peak
corresponds to the mean, median, and mode of the
distribution.
Copyright © 2011 Pearson Education, Inc.
Slide 6-4
6-C
The Normal Shape
CN (1)
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Look on p.392
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.
1. Which distribution appears to be normal?
Explain.
Copyright © 2011 Pearson Education, Inc.
Slide 6-5
6-C
Conditions for a Normal Distribution
A data set satisfying the following criteria is likely to
have a nearly normal distribution.
1. Most data values are clustered near the mean,
giving the distribution a well-defined single peak.
2. Data values are spread evenly around the mean,
making the distribution symmetric.
3. Larger deviations from the mean are increasingly
rare, producing the tapering tails of the
distribution.
4. Individual data values result from a combination
of many different factors.
Copyright © 2011 Pearson Education, Inc.
Slide 6-6
6-C
Is It a Normal Distribution?
CN (2a-b)
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2. Which of the following variables would you
expect to have a normal or nearly normal
distribution.
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a. Scores on a very easy test
b. Shoe sizes of a random sample of adult
women
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Copyright © 2011 Pearson Education, Inc.
Slide 6-7
The 68-95-99.7 Rule for
a Normal Distribution
Copyright © 2011 Pearson Education, Inc.
6-C
Slide 6-8
SAT Scores
CN (3)
6-C
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Each test that makes up the SAT is designed so
that its scores ar normally distributed with a mean
of 500 and a standard deviation of 100.
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3. Interpret this statement according to the 68-9599.7 rule
Copyright © 2011 Pearson Education, Inc.
Slide 6-9
Detecting Counterfeits
CN (4)
6-C
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Vending machines can be adjusted to reject coins
above and below certain weights. The weights of
legal US quarters are normally distributed with a
mean of 5.67 grams and a standard deviation of
.0700 gram.
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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?
Copyright © 2011 Pearson Education, Inc.
Slide 6-10
Normal Auto Prices
CN (5a-c)
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6-C
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.
5. Consider 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?
Copyright © 2011 Pearson Education, Inc.
Slide 6-11
6-C
Standard Scores
The number of standard deviations that a data
value lies above or below the mean is called its
standard score (or z-score), defined by
data value  mean
z  standard score 
standard deviation
Data Value
above the mean
below the mean
Copyright © 2011 Pearson Education, Inc.
Standard Score
positive
→
negative
→
Slide 6-12
6-C
Standard Scores
Example: If the mean were 21 with a standard
deviation of 4.7 for scores on a nationwide test, find
the z-score for a 30. What does this mean?
data value  mean
z
standard deviation
30  21

 1.91
4.7
This means that a test score of 30 would be about
1.91 standard deviations above the mean of 21.
Copyright © 2011 Pearson Education, Inc.
Slide 6-13
6-C
Standard IQ Scores
CN (6a-b)
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6. The Standford-Binet IQ test is designed so that
scores are normally distributed with a mean of
100 and a standard deviation of 16.
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Find the standard scores for IQ scores of a)95
and b)25
Copyright © 2011 Pearson Education, Inc.
Slide 6-14
6-C
Standard Scores and Percentiles
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The nth percentile of a data set is the smallest
value in the set with the property that n% of the
data are less than or equal to it.
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A data value that lies between two percentiles is
said to lie in the lower percentile.
Copyright © 2011 Pearson Education, Inc.
Slide 6-15
6-C
Standard Scores and Percentiles
Copyright © 2011 Pearson Education, Inc.
Slide 6-16
6-C
Cholesterol Levels
CN (7)
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7. Cholesterol levels in men 18 to 24 years of age
are normally distributed with a mean of 178 and a
standard deviation of 41.
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a. In what percentile is a man with cholesterol
level of 190?
b. What cholesterol level corresponds to the 90th
percentile, the level at which treatment may be
necessary?
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Copyright © 2011 Pearson Education, Inc.
Slide 6-17
6-C
Women in the Army
CN (8)
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The heights of 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 US Army, women must be between
58 inches and 80 inches tall.
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8. What percentage of women are ineligible to
serve based on their height?
Copyright © 2011 Pearson Education, Inc.
Slide 6-18
6-C
Quick Quiz
CN (9)
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9. Please go through and answer the 10 multiple
choice questions on p.398
Copyright © 2011 Pearson Education, Inc.
Slide 6-19
Homework 6C
p.398
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6-C
Discussion Paragraph for 6B
p.398:1-10
1 web
 48. SAT scores
 49. Data and Story Library
 50. Normal Distribution Demonstration
1 world
 Normal Distributions
 Non-Normal Distributions
Copyright © 2011 Pearson Education, Inc.
Slide 6-20