Estimation
(Point Estimation)
Statistical Inference for Managers
Lecture- 5
By
Imran Khan
Estimation
We are given information of a sample and using this
information, we estimate any quantity of
population.
Point Estimation:
The objective of point estimation is to obtain a
single number from the sample which will represent
the unknown value of the population parameter.
Estimator:
An estimator is a sample statistic used to estimate a
population parameter.
Properties of a good Estimator
• Unbiasedness
• Efficiency
• Consistency
Unbiasedness:
The point estimator θ^ is said to be an
unbiased estimator of θ if the expected value or
mean of the sampling distribution of θ^ is
θ.
Example: The sample mean and sample variance
are unbiased estimators of their population
parameters.
Bias
The Bias in an estimator θ^ is defined as the
difference between its mean and θ
Bias (θ^)= E(θ^)- θ
Properties of a good Estimator
Efficiency:
Is another property of a good estimate which refers
to the size of standard error of statistics. So the
most efficient estimator will be the one having
smaller standard error.
An estimator with a smaller standard error will
produce an estimate closer to the population
parameter.
Properties of a good Estimator
Consistency:
A statistic is consistent estimator of population
parameter if the value of the statistic comes very
close to the population parameter as the sample size
increases.
Consistency is a large sample property.
Properties of a good Estimator
The sample mean x̅ is the best estimator of μ as it is
un-biased, consistent and the most efficient
estimator.
Point Estimate of the population variance:
E(s²)= σ²- Proof required!
Var(x)̅= σ²/n
Estimation
Interval estimation:
An Interval estimate describes a range of values
within which a population parameter is likely to lie.
We represent confidence interval by the quantity
(1- α).
P= sample proportion
π= population proportion
Estimation
Case-1:
Interval for population mean when population
standard deviation is known:
Example
Upon collecting a sample of 250 from a population
with known standard deviation of 13.7, the mean is
found to be 112.4.
a) Find a 95% confidence interval for the mean.
b) Find a 99% confidence interval for the mean.
Case-2: Interval for Population mean when
population standard deviation is unknown and n>30:
Case-3: Interval for population mean when
population standard deviation is unknown and n<30
Using t-distribution
Example:
A business school placement officer wants to
estimate the mean annual salaries of the school’s
former students 5 years after graduation. A random
sample of 25 such graduates found a sample mean of
$42,740 and a sample standard deviation of $4,780.
Assuming that the population distribution is normal,
find a 90% confidence interval for the population
mean.
T-Table= 1.711
Case- 4: Interval for difference of two population
means when population standard deviations σ1 & σ2
are known:
Example:
The following data is given:
X1bar= 4000
X2bar= 3500
σ1= 500
n1= 16
σ2= 300
n2= 9
Find a 95% interval for the difference of two population
means?
Case-5: Interval for difference of two population
means when σ1 & σ2 are unknown and n1>30,
n2>30.
Case-6: Interval for difference of two population
means when σ1 & σ2 are unknown and n1<30,
n2<30.
Case-7: Interval for population proportion when σ is
unknown
Case-8: Interval for difference of two population
proportions
Example-1:
In a random sample of 120 large retailers, 85 used
regression as a method of forecasting. In an
independent random sample of 163 small retailers,
78 used regression as a method of forecasting. Find
a 99% confidence interval for the difference
between the two population proportions?
Example-2:
Pair
1
2
3
4
5
6
7
8
Drug-A
29
32
31
32
32
29
31
30
Drug-B
26
27
28
27
30
26
33
36
Using the above data, estimate with a 99% confidence
the mean difference in the effectiveness of the two
drugs A & B, to lower cholesterol.
Table values for Interval Estimation:
90% Confidence Interval- z= 1.64
95% Confidence Interval- z= 1.96
99% Confidence Interval- z= 2.58