Download 7-1: Graphs of Normal Probability Distributions

3/14/2013
7-1: Graphs of Normal
Probability Distributions
Important properties of a normal curve:
1. The curve is bell-shaped with the highest point over the
mean
2. The curve is symmetrical about a vertical line through the
mean
3. The curve approaches the horizontal axis but never touches
or crosses it
4. The inflection (transition) points between cupping upward
and downward occur above    and   
Diagram:
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Two Normal Curves
Both curves have the same
mean, µ = 6.
Curve A has a standard
deviation of σ = 1.
Curve B has a standard
deviation of σ = 3.
Normal Probability
• The area under any normal curve will
always be 1.
• The portion of the area under the curve
within a given interval represents the
probability that a measurement will lie in
that interval.
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Empirical rule:
For a distribution that is symmetrical and bell-shaped (in
particular, for a normal distribution):
• Approximately 68% of the data values will lie within one
standard deviation on each side of the mean.
• Approximately 95% of the data values will lie within two
standard deviations on each side of the mean.
• Approximately 99.7% (or almost all) of the data values will lie
within three standard deviations on each side of the mean.
Empirical Rule diagram:
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Empirical Rule Example
• Suppose that wheat yield per acre is
normally distributed with a mean of 35
bushels and a standard deviation of 8
bushels.
• Graph the distribution.
• Find the probability that an acre will yield
between 19 and 35 bushels.
Empirical Rule Example
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Empirical Rule Example
Examples:
A) Sketch a normal curve with a mean of 10 and standard
deviation 3.
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B) The time required to complete today’s homework assignment is
normally distributed with a mean of 30 minutes and standard
deviation of 4 minutes.
a) What percentage of students will take between 22 and 38
minutes?
b) If there are 20 students, how many will take less than 26
minutes?
c) Estimate the range of time centered about the mean in which
about 68% of the students’ homework time will be contained.
d) Estimate the probability that the assignment will not take
longer than 34 minutes.
ASSIGNMENT:
p. 253-255
#1-11
(4- a,b,e only)
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