Random Sampling Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Random Sampling.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

Random sampling is a method of selecting individuals from a population where every member has an equal chance of being chosen, ensuring the sample is unbiased and representative of the whole population.

Drawing names from a hat where all names are equally likely to be picked. No favoritism, no convenience, just pure chance. This is how we ensure the sample represents the whole population, not just the easy-to-reach people.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Random Sampling starts by matching the sample and collection method to the population named in the question.

Common stuck point: Students often know a procedure related to random sampling but skip the recognition step: Do I know the population, the sample, and the method used to choose or measure the cases? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Do I know the population, the sample, and the method used to choose or measure the cases?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Sampling Bias Mistakes

Worked Examples

Example 1

medium
A class roster has students numbered 11 to 3030. The teacher rolls a fair 3030-sided die five times and surveys those students (re-rolling duplicates). Identify the sampling method and explain why.

Answer

Simple random sampling without replacement\text{Simple random sampling without replacement}

First step

1
Each student is equally likely to be selected on any roll.

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard
Design a stratified random sample of 6060 from a population of 400400 adults with 300300 employed and 100100 unemployed, allocated proportionally.

Example 3

challenge
Design a sampling plan to estimate average monthly grocery spending in a city with 44 socio-economic strata of sizes 20,00020{,}000, 30,00030{,}000, 40,00040{,}000, 10,00010{,}000. You can afford 200200 surveys. Use proportional allocation.

Example 4

easy
A school has 500 students numbered 001–500. Describe how to select a simple random sample of 20 students using a random number generator.

Example 5

medium
Explain the difference between simple random sampling, stratified sampling, and systematic sampling. Give an example scenario where stratified sampling would be preferred.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A teacher writes all 30 student names on identical slips, mixes them in a hat, and draws 5 without looking. Is this random sampling?

Example 2

easy
A pollster surveys the first 20 people who walk out of one grocery store. Is this random sampling of the town?

Example 3

easy
To pick 4 of 100 numbered employees, a manager uses a random number generator to produce 4 distinct numbers from 1 to 100. Random sampling?

Example 4

easy
In random sampling, what probability does each population member have of being selected (for a simple random sample)?

Example 5

easy
A survey lets only people who choose to call in respond. Is the resulting sample a random sample of all customers?

Example 6

easy
A researcher numbers 50 patients and uses a random draw to pick 10. One picked patient is unavailable, so she just picks the next person on her ward instead. Did she preserve random sampling?

Example 7

easy
Which best describes the population in: 'Estimate average sleep of all students at a university by sampling 200 of them'?

Example 8

easy
A lottery machine mixes numbered balls and ejects 6. Is the set of 6 balls a random sample of the numbered balls?

Example 9

medium
A school has 600 students in three grades (200 each). A researcher wants 60 students and uses a random number generator to pick 60 from all 600 IDs. Each grade ends up with a different count. Is this still a valid random sample?

Example 10

medium
A manager wants a random sample of 5 from 200 employees but only has a list of the day-shift (120 of them). She randomly picks 5 from that list. What is wrong, and what is the actual population sampled?

Example 11

medium
Two methods to sample 30 of 300 club members: (A) put all 300 names in a drum, mix, draw 30; (B) take the 30 members whose surnames start with A-C. Which is random sampling and why?

Example 12

medium
A pollster claims a 'random sample' was obtained by surveying every visitor to a website who clicked a banner ad. Identify the type of sampling and one reason it is not random.

Example 13

medium
To survey 100 of 1000 customers, an analyst sorts IDs 1-1000 and picks every 10th (10, 20, 30, ...). Each customer's chance of inclusion is the same. Is this random sampling, and what is the chance of being selected?

Example 14

medium
A researcher uses random sampling to estimate average household income in a city of 50,000 homes, sampling 500. A colleague says 'random sampling guarantees the sample mean equals the population mean.' Is the colleague right?

Example 15

medium
A teacher wants a random sample of 6 from 24 students. He shuffles a deck mapping 24 cards to students and deals 6. A student asks: does shuffling once give each student an equal chance? Justify.

Example 16

medium
A firm samples 'randomly' by asking employees to volunteer for a 7am focus group. Only early risers volunteer. Name the bias and explain why random sampling fails.

Example 17

medium
A pollster correctly draws a simple random sample of 1000 voters but 600 refuse to answer. Does the final responding group remain a random sample, and what threat appears?

Example 18

challenge
A city has 40,000 residents: 30,000 with landlines and 10,000 without. A pollster randomly dials listed landline numbers. Every landline household is equally likely to be called. Is this a random sample of all residents? Explain the subtle issue.

Example 19

challenge
A researcher wants each of 1000 customers to have the same selection chance but wants exactly 100 selected. She assigns each a uniform random number in [0,1] and takes the 100 smallest. Does every customer have an equal chance of being in the sample? Justify.

Example 20

challenge
Pollster A samples 100% of one small neighborhood; Pollster B takes a true random sample of 500 from the whole city of 200,000. To estimate citywide opinion, which design is sounder, and what statistical idea explains why a smaller random sample beats a large convenience census of one area?

Example 21

easy
A library wants to know how often its members visit. It surveys 5050 members chosen by a random number generator from the full member roster. Is this random sampling?

Example 22

easy
A mall surveys every 1010th shopper who walks through the main entrance. What sampling method is this?

Example 23

easy
A researcher wants to study high-school students. He stands outside one school during lunch and surveys whoever stops. What is wrong with calling this a random sample of high-school students?

Example 24

easy
True or false: a sample of size 3030 chosen using a random number generator from 10001000 employees is a simple random sample.

Example 25

easy
A school splits its 400400 students into grades 9,10,11,129, 10, 11, 12 and randomly samples 2020 from each grade. What sampling method is being used?

Example 26

easy
Why is random sampling preferred over convenience sampling when estimating a population mean?

Example 27

medium
A district has 44 schools with 500,700,300,500500, 700, 300, 500 students respectively. To stratify by school, how many students should be sampled from each school for a proportional sample of 100100?

Example 28

medium
A researcher wants to study a city's neighborhoods. He randomly picks 55 of the city's 4040 neighborhoods and surveys every adult in those 55. What sampling method is this?

Example 29

medium
A pollster surveys 10001000 random landline numbers in 2025. The population of interest is 'all adults in the country.' Is this an unbiased random sample of that population?

Example 30

medium
A teacher uses a random number generator to pick 44 of 2020 groups in a class, then surveys every student in those 44 groups. Random sample of individual students?

Example 31

medium
Identify the population and sample: 'A factory inspects 2020 randomly chosen bolts from a batch of 50005000 to estimate the defect rate.'

Example 32

medium
Why does a larger random sample tend to give a more reliable estimate of a population mean than a smaller one?

Example 33

hard
A researcher has a list of 10001000 households. She picks a random starting number 11 to 1010, then takes every 1010th household after that. What sampling method, and what bias risk should she check?

Example 34

hard
A polling firm randomly picks 200200 phone numbers, then surveys only the first person from each who answers. Discuss whether the realized sample is a random sample of household members.

Example 35

hard
Two designs to study customer satisfaction at a chain with 2020 stores: (A) randomly pick 55 stores and survey all customers; (B) randomly pick 5050 customers from a master list. Which is cluster sampling, and which is simple random?

Example 36

hard
A poll randomly picks 500500 people from a city's voter registration list to estimate support for a referendum among 'all city residents.' What population is the sample technically representative of, and why might it differ from 'all residents'?

Example 37

hard
A student lab measures the heights of 3030 random classmates and reports a confidence interval. Why is calling this a confidence interval for 'all teenagers worldwide' incorrect?

Example 38

challenge
Suppose a simple random sample of nn is drawn without replacement from NN individuals. Show that the probability that two specific named individuals A and B both end up in the sample equals n(n−1)N(N−1)\frac{n(n-1)}{N(N-1)}.

Example 39

challenge
A researcher claims a 'random sample' was obtained by posting an online form on social media. Identify two distinct biases in this method.

Example 40

medium
A factory produces 10,000 widgets per day. A quality inspector wants to check 100 widgets. She takes every 100th widget off the production line, starting with widget number 37 (chosen randomly). What type of sampling is this? What potential problem could arise?

Example 41

hard
A university has 12,000 students: 7,200 undergraduates and 4,800 postgraduates. A researcher wants a stratified sample of 200 students. (a) How many undergraduates and postgraduates should be in the sample? (b) How does this compare to what might happen with a simple random sample?
← Back to Random Sampling Common Mistakes Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

stat sampling biaspopulation vs sample