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Section 7.1 – Properties of the Normal Distribution
Probability Density Function
– An equation used to compute probabilities of continuous random
variables.
It must satisfy the following 2 properties:
1.
Every point on the curve has a vertical height greater than 0.
2.
Total area under the curve must equal 1
***The area under the curve = Probability***
Normal Distribution
– A continuous random variable is normally distributed if it is symmetric and
bellshaped.
Properties of the Normal Density Curve:
1.
Symmetric about the mean,
μ.
2.
Mean = Median = Mode, so there is a single peak and the highest point occurs at
x
=
μ.
3.
Inflection points occur at
μ
–
σ
and
μ
+
σ.
4.
The area under the curve = 1.
5.
The areas to the right and left of
μ
are both equal to ½.
6.
As x increases and decreases without bound, the graph approaches (but never touches)
the horizontal axis.
7.
The Empirical Rule applies:
Approximately 68% of the area under the curve is between
μ
–
σ
and
μ
+
σ
Approximately 95% of the area under the curve is between
μ
– 2
σ
and
μ
+ 2
σ
Approximately 99.7% of the area under the curve is between
μ
– 3
σ
and
μ
+ 3
σ
(see pg. 376)
1
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View Full Document The normal curve is a mathematical model (an equation, table, or graph that is used to describe reality).
The normal curve does a good job of describing the distribution of things like height, IQ scores, birth
weights, etc.
Example
:
The heights of 10yearold males are normally distributed with a mean of 55.9 inches and a
standard deviation of 5.7 inches.
a)
Draw a normal curve with the parameters labeled.
b)
Shade the region that represents the proportion of 10yearold males who are less
than 50.2 inches tall.
c)
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This note was uploaded on 01/06/2012 for the course MATH 1342 taught by Professor Lisajuliano during the Spring '12 term at Collins.
 Spring '12
 LisaJuliano
 Normal Distribution, Probability

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