Download Section 13.6 The Normal Curve

Section 13.6
The Normal
Curve
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
INB Table of Contents
Date
2.3-2
Topic
Page #
November 4, 2013 Section 13.6 Examples
58
November 4, 2013 Section 13.6 Notes
59
November 4, 2013 Negative z-chart
60
November 4, 2013 Positive z-chart
61
November 4, 2013 Test 4 Practice Test
62
November 4, 2013 Test 4 Practice Test Workspace
63
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn

Rectangular Distribution

J-shaped Distribution

Bimodal Distribution

Skewed Distribution

Normal Distribution

z-Scores
13.6-3
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Rectangular Distribution
All the observed values occur with the
same frequency.
13.6-4
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J-shaped Distribution
The frequency is either constantly
increasing or constantly decreasing.
13.6-5
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Bimodal Distribution
Two nonadjacent values occur more
frequently than any other values in a
set of data.
13.6-6
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Skewed Distribution
Has more of a “tail” on one side than
the other.
13.6-7
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Skewed Distribution
Smoothing the histograms of the
skewed distributions to form curves.
13.6-8
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Skewed Distribution
The relationship between the mean,
median, and mode for curves that are
skewed to the right and left.
13.6-9
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Normal Distribution
The most important distribution is the
normal distribution.
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Properties of a Normal Distribution

The graph of a normal distribution is
called the normal curve.

The normal curve is bell shaped and
symmetric about the mean.

In a normal distribution, the mean,
median, and mode all have the same
value and all occur at the center of the
distribution.
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Empirical Rule

Approximately 68% of all the data lie
within one standard deviation of the
mean (in both directions).

Approximately 95% of all the data lie
within two standard deviations of the
mean (in both directions).

Approximately 99.7% of all the data lie
within three standard deviations of the
mean (in both directions).
13.6-12
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z-Scores
z-scores (or standard scores)
determine how far, in terms of
standard deviations, a given score is
from the mean of the distribution.
13.6-13
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z-Scores
The formula for finding z-scores (or
standard scores) is
value of piece of data  mean
z
standard deviation
x


13.6-14
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Example 2: Finding z-scores
A normal distribution has a mean of 80 and a standard
deviation of 10.
Find z-scores for the following values.
a) 90
b)
95
c)
80
d)
64
13.6-15
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Example 2: Finding z-scores
a) 90
13.6-16
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Example 2: Finding z-scores
b) 95
13.6-17
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Example 2: Finding z-scores
c) 80
13.6-18
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Example 2: Finding z-scores
d) 64
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To Determine the Percent of Data
Between any Two Values
1.
Draw a diagram of the normal curve
indicating the area or percent to be
determined.
2.
Use the formula to convert the given
values to z-scores. Indicate these z-scores
on the diagram.
3.
Look up the percent that corresponds to
each z-score in Table 13.7.
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To Determine the Percent of Data
Between any Two Values
a) When finding the percent of data to
the left of a negative z-score, use
Table 13.7(a).
13.6-21
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To Determine the Percent of Data
Between any Two Values
b) When finding the percent of data to
the left of a positive z-score, use
Table 13.7(b).
13.6-22
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To Determine the Percent of Data
Between any Two Values
c) When finding the percent of data to
the right of a z-score, subtract the
percent of data to the left of that zscore from 100%.
13.6-23
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To Determine the Percent of Data
Between any Two Values
c) Or use the symmetry of a normal
distribution.
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To Determine the Percent of Data
Between any Two Values
d) When finding the percent of data
between two z-scores, subtract the
smaller percent from the larger
percent.
13.6-25
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To Determine the Percent of Data
Between any Two Values
4. Change the areas you found in Step
3 to percents as explained earlier.
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Example 5: Horseback Rides
Assume that the length of time for a horseback ride on
the trail at Triple R Ranch is normally distributed with a
mean of 3.2 hours and a standard deviation of 0.4 hour.
a) What percent of horseback rides last at least 3.2
hours?
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Example 5: Horseback Rides
b) What percent of horseback rides last less than
2.8 hours?
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Example 5: Horseback Rides
c) What percent of horseback rides are at least
3.7 hours?
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Example 5: Horseback Rides
d) What percent of horseback rides are between
2.8 hours and 4.0 hours?
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Example 5: Horseback Rides
e) In a random sample of 500 horseback rides at
Triple R Ranch, how many are at least 3.7
hours?
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