Confidence Intervals
Lecture XXI
I.
Interval Estimation
A.
As we discussed when we talked about continuous distribution functions, the
probability of a specific number under a continuous distribution is zero.
B.
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.
C.
Thus, usually talk about estimated values in terms of confidence intervals.
Specifically, as in the case when we discussed the probability of a continuous
variable, we define some range of outcomes.
However, this time we usually work
the other way around defining a certain confidence level and then stating the
values that contain this confidence interval.
II.
Confidence Intervals
A.
Amemiya notes a difference between confidence and probability.
Most troubling
is our classic definition of probability as “a probabilistic statement involving
parameters.” This is troublesome due to our inability without some add
itional
Bayesian structure to state anything concrete about probabilities.
A.
Example 8.2.1.
Let
i
X
be distributed as a Bernoulli distribution,
1,2,
i
n
.
Then,
~
,
(1
) /
A
T
X
N
p p
p
n
Therefore, we have
~
0,1
1
A
T
p
Z
N
p
p
n
1.
Why?
By the Central Limit Theory
2.
Given this distribution, we can ask questions about the probability.
Specifically, we know that if
Z
is distributed
0,1
N
, then we can define
k
P
Z
k
Building on the normal probability, the one tailed probabilities for the
normal distribution are:
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AEB 6933
–
Mathematical Statistics for Food and Resource Economics
Lecture XXI
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 Fall '09
 CARRIKER
 Normal Distribution, Professor Charles Moss, Economics Professor Charles, Resource Economics Professor, Lecture XXI Fall

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