Investigating Science 11–12 · Year 12

Sample size and sample selection: drawing beads from a bag

Module 7: Fact or Fallacy?

Practical, model not builtLow risk

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The idea

Small samples scatter widely around the true proportion and the spread shrinks with the square root of sample size, while a biased way of choosing the sample misleads at every size.

What you need

a bag of 1000 beads, 300 red and 700 white (the true proportion 0.30 is kept from learners until the end)
scoops or counted draws for samples of 10, 50 and 200 beads, tally sheets, spreadsheet

How to do it

Mix the bag thoroughly; draw 10 samples of 10 beads, returning and mixing between draws; record the proportion red.
Draw 10 samples of 50 and 5 samples of 200 the same way.
Draw dot plots of the proportions for each sample size and calculate the spread.
Layer the red beads on top without mixing and take a sample of 50 from the top to model a biased selection.
Compare the spreads with √(p(1 − p)/n) and explain how sample size and selection affect a claim.

What you should see

The standard deviation of the sample proportion is √(0.3 × 0.7/n): 0.145 for n = 10, 0.065 for n = 50 and 0.032 for n = 200, so 95 % of samples fall within ± 0.28, ± 0.13 and ± 0.06 of 0.30. For samples of 10, 2.8 % hold no red beads and 98.9 % hold six or fewer; samples of 200 cluster tightly. A top-layer sample of 50 from the unmixed bag is all or nearly all red, far from 0.30, showing that a larger sample does not cure a biased selection.

What changes

What you change: sample size (and mixed or unmixed selection)
What you measure: proportion of red beads in the sample
What you keep the same: same bag
thorough mixing between draws
beads returned after each sample

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

A sample of any size represents the population.
Doubling the sample halves the error.
A large sample is always an unbiased one.

Safety card

Low riskLearners carry it out

Hazards

spilt beads are a slip hazard

Controls

draw over a tray

Note

No hazardous chemicals or heat sources: record the activity in the school's RiskAssess risk assessment, following the NSW Department of Education Science safety and compliance page; the Chemical Safety in Schools package is not triggered.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Investigating Science Stage 6 Syllabus (2017), NESA; currentINS12-14 INS11/12-4 INS11/12-5 INS11/12-6 INS11/12-7
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities