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Introduction to Business Statistics
Lecture 15
Small Sample Inference
Background:
In some occasions, the sampling cost per item is rather
high and thus one can only afford a small sample (
n
≤
30). This
presents a new challenge for statistical inference because we can no
longer approximate the sampling distributions by normal distributions.
As we knew, Central Limit Theorem requires the sample size to be
reasonably large.
The
t
statistic:
t
x
sn
=
− μ
/
,
where
s
is the sample standard deviation.
When
n
small (
≤
30),
t
statistic has a distribution different from the
standard normal one. It is called
t
 distribution.
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t
distribution:
•
Assumption: the observations are normally distributed
•
Degrees of freedom
=
−
n
1
•
It is bellshaped just like the normal distribution
•
It is more variable than normal distributions
•
The Table: Can be used to find critical values
•
Robustness: As long as the population distribution of observations
is moundshaped, the
t
statistic has nearly the same
t
distribution as
that under the normal assumption.
3
Small Sample Confidence Interval for
μ
of confidence level
)
1
(
α
−
:
xt
s
n
±
/2
Small Sample Test for
H
0
:=
0
•
OneSided Alternative:
H
a
:
>
0
or
H
a
:
<
0
•
TwoSided Alternative:
H
a
:
≠
0
•
Test Statistic:
t
x
sn
=
− μ
0
/
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Rejection Regions:
O
n
e

T
a
i
l
e
d
H
a
:
μ
>
0
H
a
:
<
0
}

{
α
t
t
t
>
}

{
t
t
t
−
<
Two Tailed
H
a
:
≠
0
}
or

{
2
/
2
/
t
t
t
t
t
−
<
>
5
Example 1.
A very costly experiment has been conducted to evaluate a
new process for producing synthetic diamonds. Six diamonds have been
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This note was uploaded on 11/23/2010 for the course BBA ISOM111 taught by Professor Hu during the Fall '08 term at HKUST.
 Fall '08
 HU
 Business

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