ECON 206
November 29, 2010
METU Department of Economics
Instructor: H. Ozan ERUYGUR
email: [email protected]
1
LECTURE 06
CONFIDENCE INTERVALS
Outline of today’s lecture:
I. Introduction
........................................................................................................................
1
II. Large Sample Confidence Intervals
..................................................................................
2
A. Interpretation of (1
α
) Confidence Intervals
.................................................................
7
B. Simulation to Show that a Confidence Interval is a Random Variable
.........................
8
III. Selecting the Sample Size
................................................................................................
9
IV. Small Sample Confidence Intervals
...............................................................................
11
A. Confidence Interval for
μ
............................................................................................
11
B. Confidence Interval for
μ
1

μ
2
......................................................................................
12
V. Confidence Intervals for
σ
2
.............................................................................................
18
Appendix Constructing Confidence interval for the
mean
..................................................
20
I. Introduction
Confidence intervals (CI) is an important part of statistical
inference.
It refers to obtaining statements such as
[
]
1
1
(
,...,
)
(
,...,
)
1
n
n
P a X
X
b X
X
θ
α
≤
≤
=
−
where
θ
is the parameter of interest and
a
,
b
are quantities
computed based on the
iid
sample
X
1
, . . . ,X
n
. The probability
1
α
−
is called the
confidence coefficient
. It is generally taken to be 0.9,
0.95 or 0.99.
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ECON 206
November 29, 2010
METU Department of Economics
Instructor: H. Ozan ERUYGUR
email: [email protected]
2
In contrast to point estimators
ˆ
θ
which give us a specific guess for
θ
, CIs provide an interval  which is less accurate than a specific
number. The advantage of confidence intervals is that we can
characterize the confidence of our statement
[
]
,
a b
θ
∈
. CIs of the
form
[
]
,
b
−∞
or
[
]
,
a
∞
are called onesided CIs (lower or upper).
•
In general, to construct a CI, we need to know some partial
information concerning the unknown distribution  for example
that it is a normal distribution.
o
Such CIs are called
small sample confidence intervals
.
•
If we can not make such an assumption we can still construct CIs
by appealing to the
central limit theorem
. However, in this case,
the CI will be only approximately correct  with the
approximation improving in its quality as the sample size
increases
n
→ ∞
.
o
Such CIs are called
large sample
confidence intervals
.
II. Large Sample Confidence Intervals
If the target parameter
θ
is
μ
or
μ
1

μ
2
, then for large samples
ECON 206
November 29, 2010
METU Department of Economics
Instructor: H. Ozan ERUYGUR
email: [email protected]
3
ˆ
ˆ
Z
θ
θ
θ
σ
−
=
possesses
approximately
a
standard
normal
distribution.
Consequently,
ˆ
ˆ
Z
θ
θ
θ
σ
−
=
forms (at least approximately) a pivotal
quantity, and hence the pivotal method can be employed to develop
intervals for the target parameter
θ
.
Example 1
Let
ˆ
θ
be a statistic that is normally distributed with
mean
θ
and standard error
ˆ
θ
σ
. Find a confidence interval for
θ
that
possesses a confidence coefficient equal to (1
α
).
Solution
We know that
ˆ
ˆ
Z
θ
θ
θ
σ
−
=
has a standard normal distribution.
Now, we select two values in the tails of this distribution, namely,
z
α
/2
and z
α
/2
, such that (see figure below)
[
]
/2
/2
1
P
z
Z
z
α
α
α
−
≤
≤
=
−
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ECON 206
November 29, 2010
METU Department of Economics
Instructor: H. Ozan ERUYGUR
email: [email protected]
4
Figure
1
Location of 
z
α
/2
and
z
α
/2
Substituting for
Z
in the probability statement yields
/2
/2
ˆ
ˆ
1
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 Spring '12
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 Economics, Normal Distribution, ........., METU Department of Economics, H. Ozan ERUYGUR

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