Sampling Distribution of Sample Variance
•
Consider a random sample of
n
observations drawn from a
population with unknown mean
µ
and unknown variance
σ
2
, when
x
1
, x
2
, …,x
n
are sample members
?
2
=
1
(
± −
1)
(
²
³
− ²̅
)
2
´
³=1
is called sample variance and the square root,
s
, is called
sample standard deviation.
•
Different random samples have different sample variance.
•
The Expected value of the sample variance is the
population variance
µ
[
?
2
] =
¶
2
•
If we assume the underlying population distribution is
normal, then it can be shown that the sample variance and
the population variance are related through a probability
distribution known as
chisquare
distribution
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Chisquare
distribution of sample and population variance
using random sample of size
n
observations from a
normally distributed population whose variance is
?
2
and
resulting sampling variance is
±
2
, it can be shown that
²
(
³−1
)
2
=
(
´ µ
1)
±
2
?
2
=
∑
(
¶
·
µ ¶̅
)
2
?
2
had a distribution known as chisquare (
²
(
³−1
)
2
)
with (n – 1)
degrees of freedom.
•
The distribution is asymmetric and defined for only
positive values since variance are all positive values.
•
We can characterize a particular member of the family of
the chisquare distributions by a single parameter referred
to as degree of freedom, denoted as
v
.
•
A chisquare distribution with
v
degrees of freedom is
denoted
²
¸
2
.
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 Fall '08
 GANDHI
 Normal Distribution, µ, σ, 1 degrees, 2 90%

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