Chapters 5 & 17: Sampling Distribution of Sample Mean and Quality
Control
Readings: Section 5.2 and Chapter 17
September 10, 2009
1
Sampling Distribution of Sample Mean
Sampling Distribution
•
Parameter: describes population
•
Statistic: describes the sample; sampling variability
•
Sampling distribution
–
A probability distribution that characterize the sampling variability
–
The distribution of values taken by a sample statistic, e.g., sample mean, across all
possible samples of the same size from the population
•
The sample mean ¯
x
is most often used as an estimate of the population mean
μ
.
–
For a particular sample, sample mean ¯
x
is a fixed number
–
Imagine taking repeated samples of size
n
from the population and calculating sample
mean for each of them: ¯
x
1
,
¯
x
2
, . . .
–
How are these sample means distributed? How closely to
μ
is ¯
x
?
Sampling Distribution of
¯
X
=
X
1
+
X
2
+
...
+
X
n
n
•
If
X
follows a normal distribution with mean
μ
and standard deviation
σ
, then
¯
X
is also
normally distributed with mean
μ
¯
X
=
μ
and standard deviation
σ
¯
X
=
σ
√
n
•
What if
X
is not normally distributed?
–
When randomly sampling from any population with mean
μ
and standard deviation
σ
, when
n
is large enough (
>
30), the sampling distribution of
¯
X
is
approximately
normal:
¯
X
∼
N
(
μ,
σ
√
n
)
–
Formula to use for
Z
:
Z
=
¯
X

μ
¯
X
σ
¯
X
=
¯
X

μ
σ/
√
n
1
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Central Limit Theorem (CLT)
Example 1
: In a certain population of fish, the lengths of individual fish follow a normal
distribution with mean 54mm and standard deviation 4.5mm.
a. What is the probability that a randomly chosen fish is between 51mm and 60mm
long?
b. Suppose we sample 4 fish. What is the mean and standard deviation of the mean
length of these 4 fish?
c. What is the probability that the mean length of the 4 fish is between 51mm and
60mm?
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 Spring '08
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 Normal Distribution, Standard Deviation, Global Warming, µ. ¯ –

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