Interval Estimation

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

Interval estimation is where an interval is found which is expected to include the unknown parameter with a specific probability.

P (t₁ where (t₁, t₂) are known as confidence interval, t₁ and t₂ are called confidence limits and k is known as confidence coefficient of the interval.

Case 1: Confidence interval for mean with unknown S.D σ

In this, the sampling from a normal population N (µ, σ2), the statistic

t=  where S²= follows t distribution with (n – 1) degree of freedom.

For 95% C.I. for mean µ,

Therefore,  is called 95% C.I. for µ.

Also,  is known as 99% C.I. for µ.

Case 2: Confidence interval for mean with known S.D.

Let us consider a random sample of size n from a Normal Population N (µ, σ2) in which cr₂ is known.

In order to find out C.I for mean µ, z= which follows standard normal distribution and 95% of area under standard normal curve lies between, z = 1.96 and z = -1.96

Therefore, P

That is, in 95% cases, it is seen that

is called 95% C.I. for µ.

Similarly,   is known as 99% C.I for µ.

Also,  is known as 99.73% C.I. for µ.

Case 3: C.I for variance cr₂ with unknown mean

In this,follows chi-square distribution with (n – 1) degrees of freedom.

For 95% probability, it is

which is 95% C.I for

Also,  which is 99% C.I for

Case 4: Confidence Interval for variance  with a known mean.

It is known that follows chi-square distribution with n degrees of freedom.

For a probability of 95%, it is observed that

that is 95% C.I for .

In the same manner, which is 99% C.I for cr.

Some C.I. are as follows:

(With Normal Population N (µ, cr2))

Difference of Means (µ ₁ – µ₂): (S.D is known).

95% Confidence limits = ( x₁- X₂ ) ± 1.96

99% confidence limits= ( x₁- X₂ ) ± 2.58

95% confidence limits= (

99% confidence limits= (

For Proportion P:

95% Confidence limits = p ± 1.96 (S.E. of p)

99% Confidence limits = p ± 2.58 (S.E. of p)

S.E of p=

For Difference of Proportions P₁- P₂:

95% Confidence limits = [(p₁ – p₂) ± 1.96 [S.E. of (p₁ – p ₂)]

99% Confidence limits = [(p₁ – p₂) ± 2.58 [S.E. of (p₁ – p₂)]

S.E. of (p₁ – p₂)=

Example 1:

A random sample of size 10 was drawn from a normal population with an unknown mean and a variance of 35.4 (emF, if the observations are (in ems): 55, 75, 71, 66, 73, 77. 63, 67, 60 and 76, obtain 99% confidence interval for the population mean.


n=10,,= 68.3

Since the population S.D. σ is known, then 99% C.I. for µ is

= = 63.45, 73.15

Example 2:

A random sample of size 10 was drawn from a normal population which are given by 48, 56, 50, 55, 49, 45, 55, 54, 47, and 43. Find 95% confidence interval for mean  µof the population.


, x= 50.2, n=10, given

Let d= x-50, then the sample becomes →-2,6, 0, 5, -1, -5, 5, 4, -3, -7.

∑d =2, ∑d2 =190

S²== =18.96


Since, the population S.D. cr is unknown, the 95% C.I. for mean is

= = 47.09, 53.31

Example 3:

The standard deviation of a random sample of size 15 drawn from a normal population is 3.2. Calculate the 95% confidence interval for the standard deviation (a) in the population.


n = 15, sample S.D (s) = 3.2

95% Confidence interval for σ2 is

From chi-square table with 14 degrees of freedom,

X20.025 = 26.12,   X20.975 =5.63

Thus, C.I is

Example 4:

A sample of500 springs produced in a factory is taken from a large consignment and 65 are found to be defective. Estimate the assign limits in which the percentage of defectives lies.


There’s 65 defective springs in a sample of size n = 500.

The sample proportion of defective is P= 65/500 = 0.13

The limits to the percentage of defectives refer to the C.I., which can be taken as

[p – 3 (S.E. of p), p + 3 (S.E. of p)]

S.E of p=== 0.02

Therefore, the limits are [0.13- 3 (0.02), 0.13 + 3 (0.02)] [0.07, 0.19].


  1. A random variable X has a distribution with density function :

0< x <2, otherwise 0. Find the MLE of the parameter a (> 0).

  1. Consider a random sample x₁, x₂,…,xn from a normal population having mean zero. Obtain the MLE of the variance and show that it is unbiased.
  2. Find the estimates of µ and σ in the normal populations N (µ, σ2) by the method of moments.
  3. Find a 95% C.I. for the mean of a normal population with σ = 3, given the sample 2.3, – 0.2, 0.4 and – 0.9.
  4. A sample of size I 0 from a normal population produces the data 2.03, 2.02, 2.01, 2.00, 1.99, 1.98, 1.97, 1.99, 1.96 and 1.95. From the sample find the 95% C.I. for the population mean.
  5. The following random sample was obtained from a normal population : 12, 9, I 0, 14, µ , 8. Find the 95% C.I. for the population S.D. when the population mean is (i) known to be 13, (ii)
  6. 228 out of 400 voters picked at random from a large electorate said that they were going to vote for a particular candidate. Find 95% C.I. for the proportion of voters of the electorate who would in favour of the candidate.
  7. A study shows that 102 of 190 persons who saw an advertisement on a product on T. V. during a sports program and 75 of 190 other persons who saw it advertised on a variety show purchased the product. Construct a 99% confidence interval for the difference of sample proportions.
  8. A random variable X has a distribution with density function: f(x) = (α +1 )xα   0<x<1α > -1= 0,            otherwise and a random sample of size 8 produces the data: 0.2, 0.4, 0.8, 0.5, 0.7, 0.9, 0.8 and 0.9. Find the MLE of the unknown parameter a.
  9. Consider a random sample of size n from a population following Poisson distribution. Obtain the MLE of the parameter of this distribution.
  10. Consider a random sample x₁, x₂,…,xn from a population following binomial distribution having parameters n and Find the MLE of p and show that it is unbiased.
  11. Show that the estimates of the parameter of the Poisson distribution obtained by the method of maximum likelihood and the method of moments are identical.
  12. In a sample of size 10, the sample mean is 3.22 and the sample variance 1.21. Find the 95% C.I. for the population mean.
  13. A random sample of size 10 from N (µ, σ2 ) yields sample mean 4.8 and sample variance 8.64. Find 95% and 99% confidence intervals for the population mean.
  14. The marks obtained by 15 students in an examination have a mean 60 and variance 30. Find 99% confidence interval for the mean of the population of marks, assuming it to be normal.
  15. In a random sample of 300 road accidents, it was found that 114 were due to bad weather. Construct a 99% confidence interval for the corresponding true proportions.


  1. σ2 = ∑ xi2 / n
  2. µ = x, σ2 = S2
  3. [- 2.54, 3.34)
  4. [ 1.972, 2.008]
  5. (i) (1.97, 6.72], (ii) (1.35, 5.30]
  6. [0.52, 0.62]
  7. (0.02, 0.28)
  8. α = o.89oo9
  9. λ =x    
  10. p = x / n.  
  11. [2.39, 4.05]
  12. 95% C.I. [2.233, 7.367], 99% C.I. (1.616, 7.984)
  13. (55.64, 64.36)

(0.31, 0.45)


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.