Worksheets · Statistics

Sampling distributions worksheet

A sample proportion p̂ is centered on the population p with standard deviation √(p(1 − p)/n). A sample mean x̄ is centered on μ with standard deviation σ/√n. Bigger samples vary less.

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Name

Sampling Distributions

Date Period

Answer each question.

  1. A population is approximately normal with mean 70 and standard deviation 48. For a random sample of 144, find the probability that xˉ is less than 76.4, to four decimal places.
  2. Of the 6000 registered voters in a town, a proportion p=0.75 have a certain trait. A random sample of 190 voters is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?
  3. In a large population p=0.6. For a random sample of 96, find the probability that p^ is greater than 0.625, to four decimal places.
  4. A population is approximately normal with mean 75 and standard deviation 12. For a random sample of 9, find the probability that xˉ is greater than 67, to four decimal places.
  5. Of the 2500 employees of a company, a proportion p=0.27 have a certain trait. A random sample of 220 employees is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?
  6. Of the 2400 students at a large high school, a proportion p=0.3 have a certain trait. A random sample of 460 students is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?
  7. In a large population p=0.7. For a random sample of 2100, find the probability that p^ is less than 0.69, to four decimal places.
  8. Of the 12000 customers of a bank, a proportion p=0.8 have a certain trait. A random sample of 660 customers is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?
  9. Of the 1500 employees of a company, a proportion p=0.51 have a certain trait. A random sample of 90 employees is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?
  10. A population is approximately normal with mean 90 and standard deviation 20. For a random sample of 16, find the probability that xˉ is less than 102, to four decimal places.
  11. In a large population p=0.2. For a random sample of 256, find the probability that p^ is less than 0.21, to four decimal places.
  12. Of the 6000 registered voters in a town, a proportion p=0.14 have a certain trait. A random sample of 1130 voters is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?
  13. Of the 1800 students at a large high school, a proportion p=0.61 have a certain trait. A random sample of 150 students is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?
  14. A population is approximately normal with mean 190 and standard deviation 7. For a random sample of 49, find the probability that xˉ is greater than 192.5, to four decimal places.
  15. Of the 8000 customers of a bank, a proportion p=0.15 have a certain trait. A random sample of 1420 customers is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?
  16. In a large population p=0.8. For a random sample of 64, find the probability that p^ is greater than 0.84, to four decimal places.
  17. A population is approximately normal with mean 170 and standard deviation 3. For a random sample of 4, find the probability that xˉ is less than 167.75, to four decimal places.
  18. Of the 12000 customers of a bank, a proportion p=0.07 have a certain trait. A random sample of 50 customers is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?
  19. A population is approximately normal with mean 145 and standard deviation 24. For a random sample of 144, find the probability that xˉ is greater than 142.6, to four decimal places.
  20. Of the 9000 registered voters in a town, a proportion p=0.2 have a certain trait. A random sample of 450 voters is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Sampling DistributionsAnswer keyVersion 1

Date Period

Answer each question.

  1. A population is approximately normal with mean 70 and standard deviation 48. For a random sample of 144, find the probability that xˉ is less than 76.4, to four decimal places.0.9452
  2. Of the 6000 registered voters in a town, a proportion p=0.75 have a certain trait. A random sample of 190 voters is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?Yes: 190≤0.1(6000)=600, and np=142.5≥10 and n(1−p)=47.5≥10
  3. In a large population p=0.6. For a random sample of 96, find the probability that p^ is greater than 0.625, to four decimal places.0.3085
  4. A population is approximately normal with mean 75 and standard deviation 12. For a random sample of 9, find the probability that xˉ is greater than 67, to four decimal places.0.9772
  5. Of the 2500 employees of a company, a proportion p=0.27 have a certain trait. A random sample of 220 employees is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?Yes: 220≤0.1(2500)=250, and np=59.4≥10 and n(1−p)=160.6≥10
  6. Of the 2400 students at a large high school, a proportion p=0.3 have a certain trait. A random sample of 460 students is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?No: the 10% condition fails, 460>0.1(2400)=240
  7. In a large population p=0.7. For a random sample of 2100, find the probability that p^ is less than 0.69, to four decimal places.0.1587
  8. Of the 12000 customers of a bank, a proportion p=0.8 have a certain trait. A random sample of 660 customers is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?Yes: 660≤0.1(12000)=1200, and np=528≥10 and n(1−p)=132≥10
  9. Of the 1500 employees of a company, a proportion p=0.51 have a certain trait. A random sample of 90 employees is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?Yes: 90≤0.1(1500)=150, and np=45.9≥10 and n(1−p)=44.1≥10
  10. A population is approximately normal with mean 90 and standard deviation 20. For a random sample of 16, find the probability that xˉ is less than 102, to four decimal places.0.9918
  11. In a large population p=0.2. For a random sample of 256, find the probability that p^ is less than 0.21, to four decimal places.0.6554
  12. Of the 6000 registered voters in a town, a proportion p=0.14 have a certain trait. A random sample of 1130 voters is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?No: the 10% condition fails, 1130>0.1(6000)=600
  13. Of the 1800 students at a large high school, a proportion p=0.61 have a certain trait. A random sample of 150 students is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?Yes: 150≤0.1(1800)=180, and np=91.5≥10 and n(1−p)=58.5≥10
  14. A population is approximately normal with mean 190 and standard deviation 7. For a random sample of 49, find the probability that xˉ is greater than 192.5, to four decimal places.0.0062
  15. Of the 8000 customers of a bank, a proportion p=0.15 have a certain trait. A random sample of 1420 customers is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?No: the 10% condition fails, 1420>0.1(8000)=800
  16. In a large population p=0.8. For a random sample of 64, find the probability that p^ is greater than 0.84, to four decimal places.0.2119
  17. A population is approximately normal with mean 170 and standard deviation 3. For a random sample of 4, find the probability that xˉ is less than 167.75, to four decimal places.0.0668
  18. Of the 12000 customers of a bank, a proportion p=0.07 have a certain trait. A random sample of 50 customers is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?No: the Large Counts condition fails, np=3.5<10
  19. A population is approximately normal with mean 145 and standard deviation 24. For a random sample of 144, find the probability that xˉ is greater than 142.6, to four decimal places.0.8849
  20. Of the 9000 registered voters in a town, a proportion p=0.2 have a certain trait. A random sample of 450 voters is taken without replacement. Are the 10% condition and the Large Counts condition both met for p^?Yes: 450≤0.1(9000)=900, and np=90≥10 and n(1−p)=360≥10

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

A population has mean 50 and standard deviation 12. For samples of 36, find the mean and standard deviation of xˉ

  1. The mean of x̄ is the population mean, 50.
  2. Its standard deviation is 12/√36 = 2.

The answer is mean 50, standard deviation 2.

Where students go wrong

Dividing by n instead of √n. Four times the sample size halves the spread; it does not quarter it.

Also called sampling distribution of p-hat, sampling distribution of x-bar or standard error.

Questions about these worksheets

Yes. Every sampling distributions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Statistics is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the sampling distributions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Dividing by n instead of √n. Four times the sample size halves the spread; it does not quarter it.