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Confdence Intervals: Lecture XXI
Charles B. Moss
October 21, 2010
Charles B. Moss ()
Confdence Interval
October 21, 2010
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Interval Estimation
2
Confdence Intervals
Charles B. Moss ()
Confdence Interval
October 21, 2010
2 / 20
Interval Estimation
As we discussed when we talked about continuous distribution
functions, the probability of a speciFc number under a continuous
distribution is zero.
Thus, if we conceptualize any estimator, either a nonparametric
estimate of the mean or a parametric estimate of a function, the
probability of the true value equal to the estimated value is obviously
zero.
Thus, usually talk about estimated values in terms of conFdence
intervals. SpeciFcally, as in the case when we discussed the
probability of a continuous variable, we deFne some range of
outcomes. However, this time we usually work the other way around
deFning a certain conFdence level and then stating the values that
contain this conFdence interval.
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October 21, 2010
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Amemiya notes a diference between conFdence and probability. Most
troubling is our classic deFnition o± probability as ”a probabilistic
statement involving parameters.” This is troublesome due to our
inability without some additional Bayesian structure to state anything
concrete about probabilities.
Example 8.2.1.
Let
X
i
be distributed as a Bernoulli distribution,
i
=1
,
2
,
···
n
. Then,
T
=
¯
X
A
∼
N
±
p
,
p
(1
−
p
)
n
²
(1)
There±ore, we have
Z
=
T
−
p
r
p
(1
−
p
)
n
A
∼
N
(0
,
1)
(2)
Charles B. Moss ()
Confdence Interval
October 21, 2010
4 / 20
Continued
I
Why? By the Central Limit Theory.
I
Given this distribution, we can ask questions about the probability.
Specifcally, we know that iF
Z
is distributed
N
(0
,
1) , then we can
defne
γ
k
=
P
(

Z

<
k
)(
3
)
Building on the normal probability, the one tailed probabilities For the
normal distribution are presented in Table 1.
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.
 Spring '10
 Staff

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