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Limits of Point estimation
•
In point estimation, statistic is used to estimate
population
parameter. Statistic is a function on
random sample.
•
From a particular sample, value of that statistic
is used to estimate the parameter. This is not the
actual value of the parameter. The estimate was
obtained at random and could be any number.
•
Most of the times, the point estimator is a cont.
r.v., then even probability of getting actual
population parameter is 0.
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View Full Document Interval estimation
•
In stead of considering a statistic as a point estimator,
we may use
random intervals
to trap the parameter.
•
In this case, the end points of the interval are r.v.s but
we can talk about the probability that it traps the
parameter value.
Definition
: A
100(1
α
)% confidence interval
(0<
α
<1)
for a parameter
θ
is a random interval [L
1
, L
2
] such
that
).
1
(
]
[
2
1
α
θ

=
≤
≤
L
L
P
Notes
:
•
Here
θ
is not a random variable. Its value is
fixed, though unknown. The interval is random.
The meaning of the probability is that for several
choices of this interval, proportion of those
which will contain
θ
is
(1
α
).
•
From a particular sample we can read off a
particular confidence interval. Here we can say
with confidence (1
α
)100% that this interval
contains
θ
. Since here nothing remains random,
we talk of confidence rather than probability.
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View Full Document Confidence interval of mean
Assume the
population X is normal
. We find the
(1
α
)100% confidence interval for population mean
μ
when
population variance
σ
2
is known
.
We know : sample mean
f8e5
X of a random sample of
size n from X has normal distribution with mean
μ
and
Variance
σ
2
/n
.
Thus
[ ]
).
1
(
/
/
).
1
(
/


2
/
2
/
2
/
α
σ
μ

=
+
<
<

∴

=
<

n
z
X
n
z
X
P
z
n
X
P
Exercise 7.4.3
When an experiment is conducted, the following
observations on the normal random variable X, the
survival time (in months)under the new treatment,
result :
8.0
13.6
13.2
13.6
12.5 14.2
14.9
14.5
13.4
8.6
11.5
16.0
14.2
19.0
17.9
17.0
X has a s.d. of 3 months.
f8e5
x=13.88, z
0.025
=1.96, so
95% C.I. is [12.41, 15.35].
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View Full Document Impracticality of assumptions
In practice, we face 2 problems in application
of this C.I. formula and need some remedies.
•
The population is not normal
•
Population variance is unknown.
• Let X
1
,X
2
,......
,X
n
be a random sample of size n
from
any
distribution with mean
μ
and variance
σ
2
. Then for
large n
,
is
approximately
normal
with mean
μ
and variance
σ
2
/n.
Furthermore, for
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This note was uploaded on 05/14/2010 for the course MATHEMATIC mathe taught by Professor Xyz during the Spring '10 term at Birla Institute of Technology & Science.
 Spring '10
 XYZ
 Math, Limits

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