1
Sampling distributions
The probability distribution of a statistic is
called a sampling distribution.
: the sampling distribution of the mean
X
n
n
...
n
...
n
X
...
X
X
X
2
2
2
2
2
X
X
n
2
1
σ
σ
σ
σ
σ
µ
µ
µ
µ
µ
=
+
+
+
=
=
+
+
+
=
+
+
+
=
2
The Central Limit Theorem
When
n
is sufficiently large (i.e. greater
than 15), the sample mean follows
approximately a normal distribution:
~
,
X
N
n
σ
µ
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3
Example 1
An electronics company manufactures
resistors that have a mean resistance of
100
Ω
and a standard deviation of 10
Ω
.
The distribution of resistance is normal.
Find the probability that a random
sample of
n
=25 resistors will have an
average resistance less than 25
Ω
.
4
Example 2
Suppose that a random variable X has a
continuous uniform distribution
Find the distribution of the sample
mean of a random sample of size n=40.
≤
≤
=
otherwise
,
0
6
x
4
,
2
/
1
)
x
(
f
5
Sampling distribution
If we have two independent populations with means
µ
1
and
µ
2
and variance
σ
1
2
and
σ
2
2
, and if
and
are the
sample means of two independent random samples of
sized n
1
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 Spring '08
 Perevalov
 Normal Distribution, Standard Deviation, Probability theory, probability density function

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