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LECTURE 3: STATISTICAL INFERENCE
Point Estimate:
The estimator (e.g.
X
) applied to the sample gives you a
single
guess
(point estimate) about the unknown parameter (e.g. the population mean
μ
).
Interval Estimate of Confidence Interval
: It gives a
range of values
likely to
contain the unknown parameter (e.g.
μ
) at a certain
confidence level
(i.e. with
certain, usually high, probability).
What does this mean?
Suppose we repeatedly drew samples from a distribution with true population
mean
μ
. For each of these samples, suppose we constructed a 95% confidence
interval for
μ
. In 95% of these samples, the calculated confidence interval around
the sample mean would include the true population mean.
Next, we calculate
twosided confidence intervals
for a population mean. We can
do the same for any other unknown parameter of a distribution, provided that we
know the (fixed sample or asymptotic) distribution of an estimator that consistently
estimates the unknown parameter of interest.
CONFIDENCE INTERALS FOR THE MEAN
Case 1: Normal random sample – known variance:
X
i
~ i.i.d.N(
μ
,
σ
2
) where
μ
unknown,
σ
2
known.
•
We specify a
confidence level
, (1 –
α
) × 100%, where
α
is a small number
between 0 and 1, e.g.
α
= 0.01, 0.05, or 0.10, which corresponds to
confidence levels of 99%, 95%, and 90%.
•
α
is the
significance level.
•
It is natural to use
X
as our estimator of
μ
. Here we know
)
n
σ
,
μ
(
N
~
X
2
Standardizing:
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1
,
0
(
N
~
n
σ
μ
X
Z
−
=
•
Based on the chosen
α
, we can determine a value c from the standard normal
table such that:
Pr(c
≤
Z
≤
c) = 1 –
α
•
Note: if you want
α
= .05, you want each tail to have 0.025
•
For example, if
α
= .05 then:
Pr(1.96
≤
Z
≤
1.96) = .95
This value c is known as our
critical value
of the standard normal distribution.
From this, we can back out a confidence interval for the true population parameter
μ
:
Pr(c
≤
Z
≤
c) = 1 –
α
Ù
Pr
(
c
n
σ
μ
X
c
≤
−
≤
−
)
= 1 –
α
Ù
Pr
(
n
σ
c
μ
X
n
σ
c
≤
−
≤
−
)
= 1 –
α
Ù
Pr
(
n
σ
c
X
μ
n
σ
c
X
+
−
≤
−
≤
−
−
)
= 1 –
α
Ù
Pr
(
n
c
X
n
c
X
σ
μ
+
≤
≤
−
)
= 1 –
α
Thus, given
α
we found an interval:
[
n
σ
c
X
,
n
σ
c
X
+
−
]
Such that the probability that
μ
lies in this interval is (1 –
α
).
NOTES:
Pr
(
n
c
X
n
c
X
σ
μ
+
≤
≤
−
)
= 1 –
α
•
Since
X
is a random variable, the endpoints of this interval are also random
variables, and the interval itself is a
random interval
. For any particular
sample, the endpoints take on specific values and the calculated confidence
interval either contains or does not contain the unknown parameter
μ
, but we
cannot know whether it does or it does not.
•
The confidence interval is larger the larger is the variance of the population
at hand
σ
, and/or the smaller is the sample sizes n, and/or the greater is the
desired confidence level (or equivalently, the smaller
α
is).
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This note was uploaded on 03/20/2009 for the course ECON 103 taught by Professor Sandrablack during the Winter '07 term at UCLA.
 Winter '07
 SandraBlack

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