POINT ESTIMATION
AND
INTERVAL ESTIMATION
DEFINITIONS
An estimator of a population parameter is a random variable that depends on the sample information and
whose realizations provide approximations to this unknown parameter. A Spescific realization of that
random variable is called an estimate.
A point estimator of a population parameter is a function of the sample information that yields a single
number. The corresponding realization is called the point estimate of the parameter.
DEFINITIONS
POPULATION PARAMETER
Mean (
ESTIMATOR

X
)
i 1
Variance (  )
Proportion (
Xi
x
n
2
(
X

X
)
i 1 i
n
2
StandartDeviation (
n
ESTIMATE
S2 

P)
)
S
n 1

s2
n
2
(
X

X
)
i
i 1
n 1
X
ˆ
P
n
s
p̂
PROPERTIES OF GOOD POINT ESTIMATORS
A good estimator must satisfy three conditions:
Unbiased: The estimator ˆ is said to be an unbiased estimator of the parameter  if the mean
of the sampling distribution of ˆ is  . In the other words the expected value of the estimator
must be equal to the mean of the parameter
E (ˆ)  
UNBIASEDNESS OF SOME ESTIMATORS
The sample mean, variance and proportion are unbiased estimators of the corresponding
population quantities.
In general, the sample standart deviation is not an unbiased estimator of the population
standart deviation.
Let ˆ be an estimator of
that is
 . The bias in ˆ
is defined as the difference between its mean and
Bias (ˆ)  E (ˆ)  
It follows that the bias of an unbiased estimator is 0.

;
EFFICIENCY
Let ˆ1 and ˆ2 be two unbiased estimators of  ,based on the same number of sample
observations. Then
(i)
ˆ1
is said to be more efficient than
ˆ2
if
Var (ˆ1 )  Var (ˆ2 )
(ii) The relative efficiency of one estimator with respect to the other is the ratio of their variances; that
is
Relative efficiency=
Var (ˆ2 )
Var (ˆ1 )
EFFICIENCY
ˆ1
is the more efficient
estimator.
If ˆ is an unbiased estimator of  , and no other unbiased estimator has smaller variance, then
most efficient or minimum variance unbiased estimator of  .
ˆ is said to be
CHOICE OF POINT ESTIMATOR
There are estimation problems for which no unbiased estimator is very satisfactory and for
which there may be much to be gained from the sacrifice of accepting little bias. One measure
of the expected closeness of an estimator ˆ to a parameter  is its mean squared error –
the expectation of the squared difference between the estimator and the parameter, that is


2

ˆ
ˆ
MSE ( )  E    


It can be shown that,

2

ˆ
ˆ
ˆ
MSE ( )  Var ( )  Bias  


CONSISTENCY
Consistency also desirable is that an estimate tend to lie nearer the population characteristic as
the sample size becomes larger. This is the basis of the property of consistency.
An estimator is a consistent estimator of a population characteristic  if the larger the sample
size, the more likely it is that the estimate will be close to  .
INTERVAL ESTIMATION
An interval estimator for apopulation parameter is a rule for determining (based on sample
information) a range, or interval, in which the parameter is likely to fall. The corresponding
estimate is called an interval estimate.
Let  be an unknown parameter. Suppose that on the basis of sample information, we can
find random variables A and B such that
P( A    B)  1  
If the specific sample realizations of A and B are denoted by a and b ,then the interval from a to
b is called a 100(1-α)% confidence interval for  . The quantity (1   ) is called the probability
content or level of confidence, of the interval.
If the population was repeatedly sampled a very large number of times, the parameter  would
be contained in 100(1-α)% of intervals calculated this way.
ELEMENTS OF CONFIDENCE INTERVAL
CONFIDENCE LIMITS FOR POPULATION
MEAN
FACTORS EFFECTING INTERVAL WIDTH
CONFIDENCE INTERVALS

KNOWN
CONFIDENCE INTERVALS

KNOWN
CONFIDENCE INTERVALS

UNKNOWN
STUDENT’S t DISTRIBUTION
STUDENT’S t TABLE
ESTIMATION FOR FINITE POPULATIONS
When sample is large relative to population,
n/N>0,05
Use finite population correction factor;
S
X  t / 2,n 1.
n
N n
S
  X  X  t / 2,n 1.
N 1
n
N n
N 1
CONFIDENCE INTERVALS FOR THE
POPULATION PROPORTION
Assumptions;
Two Categorical Outcomes (faulty/not faulty – complex/easy),
Population Follows Binomial Distribution Normal Approximation Can Be Used if:
n·p ≥ 5
n·(1 - p) ≥ 5
Confidence Interval Estimate;
pˆ x  z / 2
pˆ x (1  pˆ x )
 p  pˆ x  z / 2
n
pˆ x (1  pˆ x )
n