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Normal distribution
The normal distribution is the most widely known and used of all distributions.
Because the
normal distribution approximates many natural phenomena so well, it has developed into a
standard of reference for many probability problems.
I.
Characteristics of the Normal distribution
•
Symmetric, bell shaped
•
Continuous
for all values of X between 
∞
and
∞
so that each conceivable interval of real
numbers has a probability other than zero.
•

∞
≤
X
≤
∞
•
Two parameters, μ and
σ
.
Note that the normal distribution is actually a family of
distributions, since μ and
σ
determine the shape of the distribution.
•
The rule for a normal density function is
e
2
1
=
)
,
f(x;
2
2
/2
)

(x
2
2
σ
µ
π
•
The notation N(μ,
σ
2
) means normally distributed with mean μ and variance
σ
2
.
If we say
X
∼
N(μ,
σ
2
) we mean that X is distributed N(μ,
σ
2
).
•
About 2/3 of all cases fall within one standard deviation of the mean, that is
P(μ 
σ
≤
X
≤
μ +
σ
) = .6826.
•
About 95% of cases lie within 2 standard deviations of the mean, that is
P(μ  2
σ
≤
X
≤
μ + 2
σ
) = .9544
Normal distribution  Page 1
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View Full DocumentII.
Why is the normal distribution useful?
•
Many things actually are normally distributed, or very close to it.
For example, height
and intelligence are approximately normally distributed; measurement errors also often
have a normal distribution
•
The normal distribution is easy to work with mathematically.
In many practical cases,
the methods developed using normal theory work quite well even when the distribution is
not normal.
•
There is a very strong connection between the size of a sample N and the extent to which
a sampling distribution approaches the normal form.
Many sampling distributions based
on large N can be approximated by the normal distribution even though the population
distribution itself is definitely not normal.
III.
The standardized normal distribution.
a.
General Procedure
.
As you might suspect from the formula for the normal
density function, it would be difficult and tedious to do the calculus every time we had a new set
of parameters for
µ
and
σ
.
So instead, we usually work with the standardized normal
distribution, where μ = 0 and
σ
= 1, i.e. N(0,1).
That is, rather than directly solve a problem
involving a normally distributed variable X with mean μ and standard deviation
σ
, an indirect
approach is used.
1.
We first convert the problem into an equivalent one dealing with a normal
variable measured in standardized deviation units, called a standardized normal variable.
To do
this, if X
∼
N(μ,
σ
5
), then
1)
N(0,

X
=
Z
~
σ
µ
2.
A table of standardized normal values (Appendix E, Table I
) can then be
used to obtain an answer in terms of the converted problem.
3.
If necessary, we can then convert back to the original units of
measurement.
To do this, simply note that, if we take the formula for Z, multiply both sides by
σ
, and then add μ to both sides, we get
+
Z
=
X
4.
The interpetation of Z values is straightforward.
Since
σ
= 1, if Z = 2, the
corresponding X value is exactly 2 standard deviations above the mean.
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 Spring '11
 jh

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