MSOR221
1 / 8
Topic Two
Inferences about a Population: Interval Estimation
Point and Interval Estimation
–
A
point
estimation refers to the procedure that uses the information in the sample to
arrive at a single number that is intended to be close to the true value of the target
parameter.
An
interval
estimation refers to the procedure that uses the sample
information to arrive at the lower limit and the upper limit that are intended to
enclose the parameter of interest.
In either case the actual estimation is
accomplished by using an
estimator
for the target population parameter.
–
An
estimator
θ
ˆ
is a random variable that is used to estimate an unknown parameter
θ
in the population.
It is a mathematical function that tells how to calculate the
value of an estimate based on the random variables,
X
1
,
X
2
, …,
X
n
, contained in the
sample.
An actual numerical value obtained from an estimator is called an
estimate
.
Thus, when a single value, or point, is given as an estimate of an unknown
parameter, it is called a
point
estimate; when a range (or an interval) of values is
provided, it is called an
interval
estimate.
•
For instance,
X
may be used as an estimator of
μ
, in which case
x
is an estimate
of this parameter.
Also,
S
2
may be used as an estimator of
σ
2
, in which case
s
2
is
an estimate of this parameter.
Similarly,
P
ˆ
may be used as an estimator of
p
, in
which case
p
ˆ
is an estimate of this parameter.
–
Since estimators are random variables, consequently, estimators have their
probability distributions or, more properly, sampling distributions.
That is, by
repeated sampling from population, a probability distribution of the values of the
estimates can be constructed.
Various statistical properties can thus be used to
decide which estimator is most appropriate in a given situation, which will give us
the most information at the lowest cost, which will expose us to the smallest risk,
and so forth.
The following properties are used to evaluate the goodness of
estimators.
•
Unbiaedness
–
When we are estimating an unknown population parameter, it is highly desirable
for the sampling distribution of the estimator to cluster about the target
parameter.
In other words, the mean (or expected value) of the sampling
distribution of the estimator would equal the parameter estimated; that is,
θ
θ
=
)
ˆ
(
E
.
An estimator
θ
ˆ
is an
unbiased estimator
if it satisfies this property; otherwise
θ
ˆ
is said to be
biased
.
–
When
θ
ˆ
is a biased estimator of
θ
, it may be of interest to know the extent of the
bias
, given by
θ
θ
θ

=
)
ˆ
(
)
ˆ
(
E
B
.
•
If the bias tends to be smaller when sample size
n
grows larger, i.e.
0
)
ˆ
(
lim
=
∞
→
θ
B
n
, the estimator is
asymptotically unbiased
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
MSOR221
2 / 8
•
Efficiency
–
In addition to preferring unbiasedness, the variance of the sampling distribution
of the estimator is also preferred to be as small as possible.
Given several
unbiased estimators of a given parameter and all other things being equal, the
estimator with the smallest variance is usually selected.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 patrickchu
 Normal Distribution, Standard Deviation, standard normal distribution, θ

Click to edit the document details