Need help with your homework? Look no further! Our subject experts are ready to effortlessly handle your assignments, so you can finally say goodbye to stress and hello to top grades.

Please enable JavaScript in your browser to complete this form.
Click or drag files to this area to upload. You can upload up to 3 files.
Get a response in under 15 min

The process of drawing a sample from a population and to gather information about the parameter through a reasonably close function is known as estimation. This is done in order to find out the unknown parameters which we encounter in a population and which hinders our estimation and conclusion about a population. Thus this process is used to find out the parameters. The value obtained from such a process is known as estimated value and the function is called estimator.

An estimator is supposed to possess the qualities stated below to perform well. They are:

  1. Un biasedness

A statistic t is an unbiased estimator of a parameter , if E[t] =

If not, then the estimator is biased.

There are quite a few theorems stated below which proves this.

Theorem 1:

Prove that the sample mean x is an unbiased estimator of the population mean µ


Let x₁, x₂, …, , be a simple random sample with replacement from a finite population of size N, say, X₁, X₂,… ,


Prove E(x) =µ

While drawing xi, it can be one of the population members i.e., the probability distribution of Xi can be taken as follows:















E (  = X₁*  + X₂*  +….+ *


= µ, i= 1, 2,…., n

E (  = E [( ]

= [E (

If the population is finite or the sampling is done without replacement, the same result will be obtained.

Theorem 2:

The sample variance S²=  is a biased estimator of variance σ².


Let x₁, x₂, …,  be a random sample from an infinite population with mean σ and variance σ².

Then, E (x) = µ, Variance (xi) = E (xi – µ)² = σ², where i= 1, 2, … ,n.



=  where  xi – µ and standard deviation is unaffected by change in origin.

= – µ)²

E (s²) = – µ)²

= )= σ²-

Thus, s² is a biased estimator of σ².

Also, Let S²=

E (s²) =

= σ²

= σ²

Therefore, s² is a biased estimator of σ².

Example: A population consists of 4 values 3, 7, 11, 15. Draw all possible sample of size two with replacement. Verify that the sample mean is an unbiased estimator of the population mean.


No. of samples = 42 = 16, which are as below:

(3, 3), (7, 3), (11, 3), (15, 3)

(11 , 7), (15, 7), (11 , 11), (15, 11)

(11, 15), (15, 15), (3, 7), (7, 7),

(3, 11), (7, 11), (3, 15), (7, 15)

Population mean µ= = = 9

Sampling distribution of mean


Sample mean


Frequency f



3 1 3
5 2 10
7 3 21
9 4 36
11 3 33
13 2 26
15 1 15
Total 16 144


Mean of sample = = 9

Since E ()= µ

Therefore, sample mean is an unbiased estimator of population mean.

  1. Consistency

A statistic  obtained from a random sample of size n is said to be a consistent estimator of a parameter if it converges in probability to θ as n tends to infinity.

Alternatively, If E [ ] θ and Var [ ] 0 as n ∞, then the statistic  is said to be consistent estimator of θ.


When sampling from a population N,

E )= µ and )= → 0 as n→

Therefore, sample mean is a consistent estimator of population mean.

  1. Efficiency

A parameter might comprise of more than one consistent estimator. Let T1 and T2 be two consistent estimators of a parameter θ. If Var (T₁) < Var (T₂) for all n, then T₁ is said to be more efficient than T₂ for all size.

  1. Sufficiency

Let x₁, x ₂, … , be a random sample from a population whose pmfor pdfisf (x, 8). Then T is said to be a sufficient estimator of e if  f (x₁, ). f(x₂, )…..f( , ) = g1(T , ). g₂( x1,x2,…,

Where g1(T , ) is the sampling distribution and g₂( x1,x2,…,  is independent of .

Even though sufficient estimators exists only in certain cases, but when random sampling for a normal population, the sampling mean x is a sufficient estimator of µ.


Links of Previous Main Topic:-

Links of Next Statistics Topics:-

Homework Blues?

Get expert help with homework for all subjects.

  • NPlagiarism-free work
  • NHonest Pricing
  • NMoney-back guarantee

Latest Reviews

Solved Sample Works

Accounting Homework

Corporate Accounting Sample

Biology Homework

Genetics Assignment Sample

Essay Writing Help

Business Plan Sample

Homework Help FAQs

Our Answers to Your Questions

How do I submit my homework?

Getting homework help is very simple with us. Students can either send us the homework via email or they can upload it to our online form here. For a quicker response, You can also chat with us at WhatsApp and submit homework directly. You are sure to get a response from our side within 10 minutes.

How much will my homework cost?

The cost of paying someone to do your homework varies depending on the service and the type of assignment. We have listed our standard pricing plans for popularly used writing services. For other kind of assignments, You can get a free instant quote from us using our online form.

We also accept partial payment to start working on your assignment help. You can pay the remaining amount when your task gets completed. No pressure of up-front payment. No hidden order costs.

Can I receive help with my homework anytime?

Yes, you can receive help with your homework anytime with us. Our online homework help services are available 24/7, allowing you to receive assistance with your homework anytime, anywhere.

For urgent homework requests, reach out to us through our LiveChat or WhatsApp channels and one of our friendly support agents will assist you in finding the right expert for your online homework help request immediately. With our services, you can rely on 24/7 availability and meeting deadlines.

Are online homework websites budget-friendly for students like me?

Yes, Our Online Homework Help websites are an affordable solution for you as a student. Compared to traditional tutoring services, MyHomeworkHelp prices their homework help services honestly and within the budget of college students. This makes it easier for you to receive assistance with your homework without breaking the bank.

What is your plagiarism-free policy?

At myhomeworkhelp, we take plagiarism very seriously and ensure that all solutions provided by our tutors are original and authentic. Our tutors are trained to provide custom-made solutions, tailored specifically to meet the requirements of each student. We do not provide pre-written papers. All our homeswork solutions are made from scratch, guaranteeing 100% orignal homework answers.

Additionally, we have strict plagiarism-detection tools in place to check all submissions for authenticity.

Is using an Online Homework Help Service cheating?

Using online homework help services is not equivalent to cheating. Our services are intended to support students with their homework and provide them with the resources they need to succeed academically. With the help of our online homework help services, students can receive immediate assistance with their homework from any location, at any time.

At myhomeworkhelp, we are committed to promoting academic integrity. Our tutors provide solutions that serve as guides for drafting your own work. It is not acceptable to submit someone else's work as your own, as this constitutes academic plagiarism.

Can I chat with my tutor?

Using our secure chat board, you can now chat directly with your assigned tutor. The chats are encrypted both ways to secure your privacy. This makes your contact with the tutor directly & confidentially, so you can better explain any requirements or changes if needed or just need updates.

You can't contact the experts outside of chat board platform. Sharing any personal information, including but not limited to contact information, goes against our Terms and Conditions and therefore may result in permanently blocking you from the platform. We take any personal data very seriously and we do it for the safety of our users.

Know more about chat board here.

What is your money-back guarantee policy?

It’s worth noting that our online homework help service rarely leads to disappointment among students. Our expert tutors, along with our support and quality assurance team, are dedicated to providing the best possible experience for our customers. However, if for any reason a student is unsatisfied with their homework help solution, we offer unlimited revisions until they are fully satisfied.

In the rare event that a student remains unsatisfied even after revisions, we offer a money-back guarantee. We want all of our students to feel confident and secure when they turn to us for assistance with their homework, and this guarantee is just one way that we demonstrate our commitment to providing the best possible service. If you have any concerns about our services or the quality of the work you receive, please contact us for support.

What is the expertise of the tutor assigned to do my homework?

At myhomeworkhelp, we take pride in our team of qualified and experienced tutors. All of our tutors undergo a rigorous selection process and are required to have a minimum of a master's degree in their respective fields. Additionally, they must pass a series of tests to demonstrate their proficiency and ability to deliver quality work. We believe in transparency and providing our clients with the best possible service. You can be confident in the expertise of the tutor assigned to do your homework.

What about privacy & confidentiality?

Using My Homework Help is absolutely safe. We care about your security, therefore we encrypt all personal data to make every user feel safe while using our services and we don’t share any personal information with any third parties without your permission. Your credit card information is not stored anywhere at My Homework Help, and use of PayPal relies on their secure payment networks. Your identity, payment and homework are in safe hands. You can always be certain of getting professional help and remaining anonymous, while using My Homework Help.