SlideShare a Scribd company logo
1 of 16
Download to read offline
Point estimation
MUHAMMAD BILAL
BSSS-13-13
Presentation Topic
Point Estimation
Contents
 Statistical Inference
 Estimation
A. Point Estimation
B. Estimator
C. Estimate
 Criteria of a good Estimator
A. Unbiasedness
B. Consistency
C. Efficiency
D. Sufficiency
Statistical Inference
Usually the population is not known completely we can obtain information
about population parameters by use of samples drawn from it. Statistical
inference deals with such problems.it is defined as
“The art of drawing conclusions or inferences about the unknown
parameters of the populations from the limited information contained in
the sample”
Two important parts of statistical inference are:
1. Estimation of parameter
2. Testing of Hypotheses
Estimation
It is a procedure of making judgment about the true but unknown values
of the population parameters from the sample observations
It is further divided into two parts
 Point Estimation
 Interval Estimation
Point Estimation
If we express an estimate by a single value, it is called point Estimation.
For example the value of X bar (the sample mean) computed from a
sample of size n is a point estimate of the population parameter µ
Estimator
A rule used to estimate a numerical value is called estimator.
The estimator of mean is given below
X bar= 𝑖=1
𝑛 𝑋
𝑛
Estimate
An estimate is a numerical value of the unknown parameter obtained by
applying a formula (estimator) to a particular sample.
If θ is a parameter Ô denotes it’s estimate
Example: Let a sample of size 5 be 2,4,5,9,10.then an estimate of the
population mean µ,obtained by applying an estimator
X bar= 𝑖=1
𝑛 𝑋
𝑛
Estimator
X bar=
2+4+5+9+10
5
X bar=
30
5
=6 Estimate
Criteria for good point Estimators
A point estimator is considered a good estimator if it is satisfies various criteria.
Four of criteria are
A. Unbiasedness
B. Consistency
C. Efficiency
D. Sufficiency
 Unbiasedness
The bias of an estimator Ô is defines as the difference between it’s
expected value and the true parameter θ
Bias=E(Ô) – θ
An estimator is defined to be unbiased if the statistic used an estimator has
it’s expected value equal to the true value of the population parameter
being estimated.
E(Ô)=θ
The estimator is defined to be positively biased when
E(Ô)>θ
The estimator is defined to be negatively biased when
E(Ô)<θ
 Consistency
An estimator is said to be consistent if the statistic to be used as estimator
becomes closer and closer to the population parameter being estimator as the
sample size n increases.
lim
𝑛→∞
𝑃[|Ô − θ| ≤ 𝑒] = 1
A consistent estimator may or may not be unbiased.
The sample mean X bar=
1
𝑛 𝑖=1
𝑛
𝑥 which is an unbiased estimator of µ, is a
consistent estimator of the mean µ.
To prove that an estimator is consistent, we may state a criterion that is
sometimes quite useful, as follows
“Let Ô be an estimator of θ based on a sample of size n. Then Ô is a
consistent estimator of θ,if Var(Ô) approaches 0 as n approaches
infinity.”
 Efficiency
An unbiased estimator is defined to be efficient if the variance of its
sampling distribution is smaller than that of sampling distribution of
any other any other unbiased estimator of the same parameter .IN
other words suppose there are two unbiased estimator T1 and T2 of the
same parameter θ , then T1 will be said to be more efficient estimator
than T2.If Var(T1)<Var(T2).
i. An estimator is more efficient if it is unbiased as well as has the
minimum variance as compared with any other unbiased estimator
ii. The relative efficiency of T1 compare to T2 is given by the ratio
Ef=Var(T2)/Var(T1) which is greater than 1
 Mean
Var(x bar)=
σ2
n
 Median
Var(x bar)=
𝝅
𝟐
.
σ2
n
=(1.57)
σ2
n
 Sufficiency
An estimator is defined to be sufficient if the statistic used as estimator
uses all the information that contained in the sample.
Any statistic that is not computed from all the values in the sample is not
a sufficient estimator. The sample mean X bar is a sufficient estimator of
µ.This implies that x bar contains all the information in the sample
relative to the estimation of population parameter µ and no other
estimator such as the sample median etc.
Any Questions?

More Related Content

What's hot

Lecture 5: Interval Estimation
Lecture 5: Interval Estimation Lecture 5: Interval Estimation
Lecture 5: Interval Estimation Marina Santini
 
Maximum likelihood estimation
Maximum likelihood estimationMaximum likelihood estimation
Maximum likelihood estimationzihad164
 
Statistical inference concept, procedure of hypothesis testing
Statistical inference   concept, procedure of hypothesis testingStatistical inference   concept, procedure of hypothesis testing
Statistical inference concept, procedure of hypothesis testingAmitaChaudhary19
 
Introduction to Probability and Probability Distributions
Introduction to Probability and Probability DistributionsIntroduction to Probability and Probability Distributions
Introduction to Probability and Probability DistributionsJezhabeth Villegas
 
F Distribution
F  DistributionF  Distribution
F Distributionjravish
 
Statistical Estimation
Statistical Estimation Statistical Estimation
Statistical Estimation Remyagharishs
 
Regression analysis
Regression analysisRegression analysis
Regression analysisSohag Babu
 
Skewness & Kurtosis
Skewness & KurtosisSkewness & Kurtosis
Skewness & KurtosisNavin Bafna
 
Ch4 Confidence Interval
Ch4 Confidence IntervalCh4 Confidence Interval
Ch4 Confidence IntervalFarhan Alfin
 
Introduction to Statistics - Basic concepts
Introduction to Statistics - Basic conceptsIntroduction to Statistics - Basic concepts
Introduction to Statistics - Basic conceptsDocIbrahimAbdelmonaem
 
Probability distribution
Probability distributionProbability distribution
Probability distributionRohit kumar
 
Hypothesis testing an introduction
Hypothesis testing an introductionHypothesis testing an introduction
Hypothesis testing an introductionGeetika Gulyani
 
Interval Estimation & Estimation Of Proportion
Interval Estimation & Estimation Of ProportionInterval Estimation & Estimation Of Proportion
Interval Estimation & Estimation Of ProportionDataminingTools Inc
 

What's hot (20)

Lecture 5: Interval Estimation
Lecture 5: Interval Estimation Lecture 5: Interval Estimation
Lecture 5: Interval Estimation
 
Sampling distribution
Sampling distributionSampling distribution
Sampling distribution
 
Probability concept and Probability distribution
Probability concept and Probability distributionProbability concept and Probability distribution
Probability concept and Probability distribution
 
Maximum likelihood estimation
Maximum likelihood estimationMaximum likelihood estimation
Maximum likelihood estimation
 
Methods of point estimation
Methods of point estimationMethods of point estimation
Methods of point estimation
 
Statistical inference concept, procedure of hypothesis testing
Statistical inference   concept, procedure of hypothesis testingStatistical inference   concept, procedure of hypothesis testing
Statistical inference concept, procedure of hypothesis testing
 
Introduction to Probability and Probability Distributions
Introduction to Probability and Probability DistributionsIntroduction to Probability and Probability Distributions
Introduction to Probability and Probability Distributions
 
F Distribution
F  DistributionF  Distribution
F Distribution
 
Sampling Distributions
Sampling DistributionsSampling Distributions
Sampling Distributions
 
Statistical Estimation
Statistical Estimation Statistical Estimation
Statistical Estimation
 
Regression analysis
Regression analysisRegression analysis
Regression analysis
 
STATISTIC ESTIMATION
STATISTIC ESTIMATIONSTATISTIC ESTIMATION
STATISTIC ESTIMATION
 
Skewness & Kurtosis
Skewness & KurtosisSkewness & Kurtosis
Skewness & Kurtosis
 
Ch4 Confidence Interval
Ch4 Confidence IntervalCh4 Confidence Interval
Ch4 Confidence Interval
 
Introduction to Statistics - Basic concepts
Introduction to Statistics - Basic conceptsIntroduction to Statistics - Basic concepts
Introduction to Statistics - Basic concepts
 
Bernoulli distribution
Bernoulli distributionBernoulli distribution
Bernoulli distribution
 
Probability distribution
Probability distributionProbability distribution
Probability distribution
 
Hypothesis testing an introduction
Hypothesis testing an introductionHypothesis testing an introduction
Hypothesis testing an introduction
 
Significance test
Significance testSignificance test
Significance test
 
Interval Estimation & Estimation Of Proportion
Interval Estimation & Estimation Of ProportionInterval Estimation & Estimation Of Proportion
Interval Estimation & Estimation Of Proportion
 

Similar to Point estimation

Point estimation.pptx
Point estimation.pptxPoint estimation.pptx
Point estimation.pptxDrNidhiSinha
 
statistical estimation
statistical estimationstatistical estimation
statistical estimationAmish Akbar
 
inferencial statistics
inferencial statisticsinferencial statistics
inferencial statisticsanjaemerry
 
Basic of Statistical Inference Part-III: The Theory of Estimation from Dexlab...
Basic of Statistical Inference Part-III: The Theory of Estimation from Dexlab...Basic of Statistical Inference Part-III: The Theory of Estimation from Dexlab...
Basic of Statistical Inference Part-III: The Theory of Estimation from Dexlab...Dexlab Analytics
 
Statistical inference 2
Statistical inference 2Statistical inference 2
Statistical inference 2safi Ullah
 
Elementary statistics for Food Indusrty
Elementary statistics for Food IndusrtyElementary statistics for Food Indusrty
Elementary statistics for Food IndusrtyAtcharaporn Khoomtong
 
Overview of Advance Marketing Research
Overview of Advance Marketing ResearchOverview of Advance Marketing Research
Overview of Advance Marketing ResearchEnamul Islam
 
Business statistics-i-part2-aarhus-bss
Business statistics-i-part2-aarhus-bssBusiness statistics-i-part2-aarhus-bss
Business statistics-i-part2-aarhus-bssAntonio Rivero Ostoic
 
Topic 2 Measures of Central Tendency.pptx
Topic 2   Measures of Central Tendency.pptxTopic 2   Measures of Central Tendency.pptx
Topic 2 Measures of Central Tendency.pptxCallplanetsDeveloper
 
statistical inference.pptx
statistical inference.pptxstatistical inference.pptx
statistical inference.pptxSoujanyaLk1
 
Chapter 7 sampling distributions
Chapter 7 sampling distributionsChapter 7 sampling distributions
Chapter 7 sampling distributionsmeharahutsham
 

Similar to Point estimation (20)

Point estimation.pptx
Point estimation.pptxPoint estimation.pptx
Point estimation.pptx
 
statistical estimation
statistical estimationstatistical estimation
statistical estimation
 
inferencial statistics
inferencial statisticsinferencial statistics
inferencial statistics
 
Basic of Statistical Inference Part-III: The Theory of Estimation from Dexlab...
Basic of Statistical Inference Part-III: The Theory of Estimation from Dexlab...Basic of Statistical Inference Part-III: The Theory of Estimation from Dexlab...
Basic of Statistical Inference Part-III: The Theory of Estimation from Dexlab...
 
Statistical inference 2
Statistical inference 2Statistical inference 2
Statistical inference 2
 
elementary statistic
elementary statisticelementary statistic
elementary statistic
 
Elementary statistics for Food Indusrty
Elementary statistics for Food IndusrtyElementary statistics for Food Indusrty
Elementary statistics for Food Indusrty
 
Overview of Advance Marketing Research
Overview of Advance Marketing ResearchOverview of Advance Marketing Research
Overview of Advance Marketing Research
 
Estimating a Population Proportion
Estimating a Population Proportion  Estimating a Population Proportion
Estimating a Population Proportion
 
chapter12.ppt
chapter12.pptchapter12.ppt
chapter12.ppt
 
3 es timation-of_parameters[1]
3 es timation-of_parameters[1]3 es timation-of_parameters[1]
3 es timation-of_parameters[1]
 
Sampling
SamplingSampling
Sampling
 
Business statistics-i-part2-aarhus-bss
Business statistics-i-part2-aarhus-bssBusiness statistics-i-part2-aarhus-bss
Business statistics-i-part2-aarhus-bss
 
Statistics
StatisticsStatistics
Statistics
 
Statistics
StatisticsStatistics
Statistics
 
Topic 2 Measures of Central Tendency.pptx
Topic 2   Measures of Central Tendency.pptxTopic 2   Measures of Central Tendency.pptx
Topic 2 Measures of Central Tendency.pptx
 
statistical inference.pptx
statistical inference.pptxstatistical inference.pptx
statistical inference.pptx
 
Statistics
StatisticsStatistics
Statistics
 
Statistics
StatisticsStatistics
Statistics
 
Chapter 7 sampling distributions
Chapter 7 sampling distributionsChapter 7 sampling distributions
Chapter 7 sampling distributions
 

Recently uploaded

The Singapore Teaching Practice document
The Singapore Teaching Practice documentThe Singapore Teaching Practice document
The Singapore Teaching Practice documentXsasf Sfdfasd
 
How to Use api.constrains ( ) in Odoo 17
How to Use api.constrains ( ) in Odoo 17How to Use api.constrains ( ) in Odoo 17
How to Use api.constrains ( ) in Odoo 17Celine George
 
HED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfHED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfMohonDas
 
How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17Celine George
 
CAULIFLOWER BREEDING 1 Parmar pptx
CAULIFLOWER BREEDING 1 Parmar pptxCAULIFLOWER BREEDING 1 Parmar pptx
CAULIFLOWER BREEDING 1 Parmar pptxSaurabhParmar42
 
How to Print Employee Resume in the Odoo 17
How to Print Employee Resume in the Odoo 17How to Print Employee Resume in the Odoo 17
How to Print Employee Resume in the Odoo 17Celine George
 
Ultra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxUltra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxDr. Asif Anas
 
Practical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptxPractical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptxKatherine Villaluna
 
PISA-VET launch_El Iza Mohamedou_19 March 2024.pptx
PISA-VET launch_El Iza Mohamedou_19 March 2024.pptxPISA-VET launch_El Iza Mohamedou_19 March 2024.pptx
PISA-VET launch_El Iza Mohamedou_19 March 2024.pptxEduSkills OECD
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17Celine George
 
The Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George WellsThe Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George WellsEugene Lysak
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxiammrhaywood
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17Celine George
 
Presentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphPresentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphNetziValdelomar1
 
How to Add a many2many Relational Field in Odoo 17
How to Add a many2many Relational Field in Odoo 17How to Add a many2many Relational Field in Odoo 17
How to Add a many2many Relational Field in Odoo 17Celine George
 
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptxmary850239
 
Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.raviapr7
 
Human-AI Co-Creation of Worked Examples for Programming Classes
Human-AI Co-Creation of Worked Examples for Programming ClassesHuman-AI Co-Creation of Worked Examples for Programming Classes
Human-AI Co-Creation of Worked Examples for Programming ClassesMohammad Hassany
 

Recently uploaded (20)

Personal Resilience in Project Management 2 - TV Edit 1a.pdf
Personal Resilience in Project Management 2 - TV Edit 1a.pdfPersonal Resilience in Project Management 2 - TV Edit 1a.pdf
Personal Resilience in Project Management 2 - TV Edit 1a.pdf
 
The Singapore Teaching Practice document
The Singapore Teaching Practice documentThe Singapore Teaching Practice document
The Singapore Teaching Practice document
 
How to Use api.constrains ( ) in Odoo 17
How to Use api.constrains ( ) in Odoo 17How to Use api.constrains ( ) in Odoo 17
How to Use api.constrains ( ) in Odoo 17
 
HED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfHED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdf
 
How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17
 
CAULIFLOWER BREEDING 1 Parmar pptx
CAULIFLOWER BREEDING 1 Parmar pptxCAULIFLOWER BREEDING 1 Parmar pptx
CAULIFLOWER BREEDING 1 Parmar pptx
 
How to Print Employee Resume in the Odoo 17
How to Print Employee Resume in the Odoo 17How to Print Employee Resume in the Odoo 17
How to Print Employee Resume in the Odoo 17
 
Ultra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxUltra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptx
 
Finals of Kant get Marx 2.0 : a general politics quiz
Finals of Kant get Marx 2.0 : a general politics quizFinals of Kant get Marx 2.0 : a general politics quiz
Finals of Kant get Marx 2.0 : a general politics quiz
 
Practical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptxPractical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptx
 
PISA-VET launch_El Iza Mohamedou_19 March 2024.pptx
PISA-VET launch_El Iza Mohamedou_19 March 2024.pptxPISA-VET launch_El Iza Mohamedou_19 March 2024.pptx
PISA-VET launch_El Iza Mohamedou_19 March 2024.pptx
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17
 
The Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George WellsThe Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George Wells
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17
 
Presentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphPresentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a Paragraph
 
How to Add a many2many Relational Field in Odoo 17
How to Add a many2many Relational Field in Odoo 17How to Add a many2many Relational Field in Odoo 17
How to Add a many2many Relational Field in Odoo 17
 
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx
 
Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.
 
Human-AI Co-Creation of Worked Examples for Programming Classes
Human-AI Co-Creation of Worked Examples for Programming ClassesHuman-AI Co-Creation of Worked Examples for Programming Classes
Human-AI Co-Creation of Worked Examples for Programming Classes
 

Point estimation

  • 4. Contents  Statistical Inference  Estimation A. Point Estimation B. Estimator C. Estimate  Criteria of a good Estimator A. Unbiasedness B. Consistency C. Efficiency D. Sufficiency
  • 5. Statistical Inference Usually the population is not known completely we can obtain information about population parameters by use of samples drawn from it. Statistical inference deals with such problems.it is defined as “The art of drawing conclusions or inferences about the unknown parameters of the populations from the limited information contained in the sample” Two important parts of statistical inference are: 1. Estimation of parameter 2. Testing of Hypotheses
  • 6. Estimation It is a procedure of making judgment about the true but unknown values of the population parameters from the sample observations It is further divided into two parts  Point Estimation  Interval Estimation
  • 7. Point Estimation If we express an estimate by a single value, it is called point Estimation. For example the value of X bar (the sample mean) computed from a sample of size n is a point estimate of the population parameter µ Estimator A rule used to estimate a numerical value is called estimator. The estimator of mean is given below X bar= 𝑖=1 𝑛 𝑋 𝑛
  • 8. Estimate An estimate is a numerical value of the unknown parameter obtained by applying a formula (estimator) to a particular sample. If θ is a parameter Ô denotes it’s estimate Example: Let a sample of size 5 be 2,4,5,9,10.then an estimate of the population mean µ,obtained by applying an estimator X bar= 𝑖=1 𝑛 𝑋 𝑛 Estimator X bar= 2+4+5+9+10 5 X bar= 30 5 =6 Estimate
  • 9. Criteria for good point Estimators A point estimator is considered a good estimator if it is satisfies various criteria. Four of criteria are A. Unbiasedness B. Consistency C. Efficiency D. Sufficiency
  • 10.  Unbiasedness The bias of an estimator Ô is defines as the difference between it’s expected value and the true parameter θ Bias=E(Ô) – θ An estimator is defined to be unbiased if the statistic used an estimator has it’s expected value equal to the true value of the population parameter being estimated. E(Ô)=θ The estimator is defined to be positively biased when E(Ô)>θ The estimator is defined to be negatively biased when E(Ô)<θ
  • 11.  Consistency An estimator is said to be consistent if the statistic to be used as estimator becomes closer and closer to the population parameter being estimator as the sample size n increases. lim 𝑛→∞ 𝑃[|Ô − θ| ≤ 𝑒] = 1 A consistent estimator may or may not be unbiased. The sample mean X bar= 1 𝑛 𝑖=1 𝑛 𝑥 which is an unbiased estimator of µ, is a consistent estimator of the mean µ.
  • 12. To prove that an estimator is consistent, we may state a criterion that is sometimes quite useful, as follows “Let Ô be an estimator of θ based on a sample of size n. Then Ô is a consistent estimator of θ,if Var(Ô) approaches 0 as n approaches infinity.”
  • 13.  Efficiency An unbiased estimator is defined to be efficient if the variance of its sampling distribution is smaller than that of sampling distribution of any other any other unbiased estimator of the same parameter .IN other words suppose there are two unbiased estimator T1 and T2 of the same parameter θ , then T1 will be said to be more efficient estimator than T2.If Var(T1)<Var(T2). i. An estimator is more efficient if it is unbiased as well as has the minimum variance as compared with any other unbiased estimator ii. The relative efficiency of T1 compare to T2 is given by the ratio Ef=Var(T2)/Var(T1) which is greater than 1
  • 14.  Mean Var(x bar)= σ2 n  Median Var(x bar)= 𝝅 𝟐 . σ2 n =(1.57) σ2 n
  • 15.  Sufficiency An estimator is defined to be sufficient if the statistic used as estimator uses all the information that contained in the sample. Any statistic that is not computed from all the values in the sample is not a sufficient estimator. The sample mean X bar is a sufficient estimator of µ.This implies that x bar contains all the information in the sample relative to the estimation of population parameter µ and no other estimator such as the sample median etc.