Unit 1 · Topic 1.11 Beta

Random Sampling

Random sampling uses chance to choose individuals from a sampling frame.

Practice 2: Collect Data

Part 1 · Hook

Why this matters

A principal wants 100 students’ opinions on a new schedule. She could number all 1,200 students and pick 100 by chance, pick 25 from each grade, pick four whole homerooms, or take every 12th name on a list. All four are random. They differ in cost, in convenience, and in how well they guarantee that every grade is heard.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. A school wants to know what its 1,200 students think. What is the population?

  1. The 1,200 students
  2. The 100 students asked
  3. The principal
  4. The schedule
Show the answer

The population is the whole group the question is about.

  • Correct: The 1,200 students:
  • The 100 students asked:
  • The principal:
  • The schedule:

2. A study asks 100 students their opinion without imposing anything. What kind of study is it?

  1. A sample survey
  2. An experiment
  3. A census
  4. None of these
Show the answer

It asks a sample questions, so it is a sample survey, an observational study.

  • Correct: A sample survey:
  • An experiment:
  • A census:
  • None of these:

Part 4 · See it

See it first

Four copies of the same population of 48 students in 6 homerooms of 8, with selected students in plum. Simple random sample: any 8 of the 48, scattered. Stratified random sample: the grades are strata, two homerooms each, and 3 students are chosen at random from each grade. Cluster sample: 2 whole homerooms chosen at random. Systematic random sample: a random start at the 3rd student, then every 6th student in order.
One population of 48 students, four random samples: simple random, stratified by grade, cluster (two whole homerooms) and systematic (every 6th after a random start). LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Use chance, not judgment, to choose the sampleno one’s preferences decide who is in it, so it tends to represent the population
  2. Label every individual in the sampling frame and pick labels at random, skipping repeatsyou get a simple random sample: every group of n has the same chance
  3. Split the population into strata of similar individuals and sample eachevery group is represented, and estimates vary less when strata differ
  4. Choose whole clusters, or every kth individual after a random startsampling is cheaper and easier, at some cost in precision or with a risk from patterns in the list

Part 6 · Key ideas

Key ideas

  • A simple random sample (SRS) of size n gives every group of n individuals the same chance of being chosen. Procedure: label, generate random numbers, skip repeats, select.
  • Sampling without replacement means each individual can be chosen once; with replacement allows repeats.
  • A stratified random sample splits the population into strata of similar individuals and takes an SRS from each.
  • A cluster sample chooses whole groups at random and measures everyone in them. A systematic random sample takes every kth individual after a random start.
  • The sampling frame is the list the sample is drawn from.

Part 7 · Misconception

A common mistake

The wrong idea: "Use a random number generator to choose 10 ponds" is a complete procedure.

What actually happens: Someone else could not follow it. Say how the ponds are labeled (1 to 80), what is generated (whole numbers from 1 to 80), what to do with repeats (skip them) and when to stop (10 different ponds).

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Four ways to sample a school

A high school has 1,200 students: 300 in each of grades 9, 10, 11 and 12, in 48 homerooms of 25 students. The principal wants about 100 students’ opinions on a new schedule. Four plans:

  • Plan P: number the students from 1 to 1,200 and use a random number generator to choose 100 different numbers.
  • Plan Q: choose 25 students at random from each grade.
  • Plan R: choose 4 of the 48 homerooms at random and survey every student in them.
  • Plan S: list the students alphabetically, choose a random starting point among the first 12, then take every 12th student.

1. What kind of sample does plan P give?

  1. A stratified random sample
  2. A simple random sample
  3. A cluster sample
  4. A systematic random sample, since a generator is used
Show the answer

Choosing n individuals so that every possible group of n has the same chance is a simple random sample.

  • A stratified random sample: Numbering the students is a step in choosing; nothing is split into strata.
  • Correct: A simple random sample: Right: every group of 100 students has the same chance of being the sample.
  • A cluster sample: A cluster sample takes whole groups such as homerooms; plan P picks individual students.
  • A systematic random sample, since a generator is used: A systematic sample takes every kth individual after a random start.

2. What kind of sample does plan Q give?

  1. A cluster sample, with the grades as the clusters
  2. A simple random sample of 100 of the school’s students
  3. A systematic random sample of 25 students per grade
  4. A stratified random sample, with the grades as strata
Show the answer

Splitting the population into strata and taking a simple random sample from each is a stratified random sample.

  • A cluster sample, with the grades as the clusters: In a cluster sample whole groups are chosen and everyone in them is surveyed; plan Q samples within every grade.
  • A simple random sample of 100 of the school’s students: Not every group of 100 is possible: a sample of 100 ninth graders cannot happen.
  • A systematic random sample of 25 students per grade: No every kth rule is used; students are chosen at random within each grade.
  • Correct: A stratified random sample, with the grades as strata: Right: the population is split into groups by grade and a simple random sample is taken from each.

3. What kind of sample does plan R give, and what is its main advantage?

  1. A cluster sample: whole homerooms make it easy
  2. A stratified sample: it guarantees that each grade is included
  3. A simple random sample: each student has the same chance of selection
  4. A systematic sample
Show the answer

Choosing whole groups (clusters) at random and measuring every individual in them is cluster sampling. Its strength is convenience.

  • Correct: A cluster sample: whole homerooms make it easy: Right: clusters are chosen at random and everyone in them is included, which saves time.
  • A stratified sample: it guarantees that each grade is included: Homerooms are chosen at random, so some grades may be missed; it is a cluster sample.
  • A simple random sample: each student has the same chance of selection: Each student does have the same chance, but not every group of 100 is possible, so it is not simple random.
  • A systematic sample: The 4 homerooms are chosen at random, not by an every kth rule.

4. The principal thinks opinions on the schedule differ a lot from grade to grade. Which plan best makes sure each grade’s view is represented?

  1. Plan R
  2. Plan P
  3. Plan Q
  4. Plan S
Show the answer

Stratify by a variable that is related to the response: each stratum is represented, and estimates vary less from sample to sample.

  • Plan R: Four random homerooms could all come from one or two grades.
  • Plan P: A simple random sample could by chance include few students from some grade.
  • Correct: Plan Q: Right: stratifying by grade guarantees 25 students from each grade, and when grades differ it gives more precise results.
  • Plan S: An alphabetical list has nothing to do with grade, so grades may be uneven.

Table

Testing pond water

A park has 80 ponds, listed on a park map. A wildlife team has the time and equipment to test the water in 10 of them, and wants a simple random sample of 10 ponds.

5. Put the steps for choosing the simple random sample of 10 ponds in order.

  1. Label the 80 ponds on the map with the numbers 1 to 80
  2. Use a random number generator to produce whole numbers from 1 to 80
  3. Skip any number that has already been chosen
  4. Keep going until 10 different numbers are chosen
  5. Test the water in the 10 ponds with those numbers
Show the answer

Label, generate random numbers, skip repeats, stop at 10 different labels, then measure those individuals.

  • Correct order: 1. Label the 80 ponds on the map with the numbers 1 to 80 2. Use a random number generator to produce whole numbers from 1 to 80 3. Skip any number that has already been chosen 4. Keep going until 10 different numbers are chosen 5. Test the water in the 10 ponds with those numbers

6. A team member writes, "Use a random number generator to pick 10 ponds." Why is this not enough as a procedure?

  1. It leaves out labeling 1 to 80 and handling repeats
  2. A random number generator gives the same pond each time
  3. Ten ponds is too few
  4. Ponds have to be picked from a table, not from a generator
Show the answer

A procedure must be clear enough that another person could carry it out exactly: label 1 to 80, generate, skip repeats, choose 10.

  • Correct: It leaves out labeling 1 to 80 and handling repeats: Right: someone else could not follow it. A procedure must label, generate, skip repeats and select.
  • A random number generator gives the same pond each time: A generator gives different numbers; the issue is the missing steps.
  • Ten ponds is too few: The sample size was set by time and equipment; the procedure is what is incomplete.
  • Ponds have to be picked from a table, not from a generator: Either a table or a generator works when the steps are spelled out.

7. How should strata differ from clusters?

  1. Strata are chosen at random, and clusters are chosen by the researcher
  2. Strata are larger groups of people than clusters
  3. Strata hold similar individuals; clusters each mix the population
  4. Strata and clusters are two names for the same idea
Show the answer

Stratify by groups whose members are similar to each other; cluster by groups that each resemble the whole population.

  • Strata are chosen at random, and clusters are chosen by the researcher: In stratified sampling every stratum is sampled; in cluster sampling clusters are chosen at random.
  • Strata are larger groups of people than clusters: Size does not define them; what matters is how alike the individuals are within each group.
  • Correct: Strata hold similar individuals; clusters each mix the population: Right: strata group similar individuals; clusters work best when each one mixes the population.
  • Strata and clusters are two names for the same idea: They are used in opposite ways: sample within every stratum, or survey all of a few clusters.

Part 9 · Summary

Summary

Random sampling uses chance to choose individuals from a sampling frame. A simple random sample gives every group of n the same chance; a stratified sample takes an SRS within each stratum of similar individuals; a cluster sample takes whole groups chosen at random; a systematic sample takes every kth individual after a random start. Write procedures as label, generate, skip repeats, select.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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