Population estimates and margin of error
Interpret an estimate and its uncertainty, and connect sample design and size to the strength of a conclusion.
What you will tackle
- Use sample means and proportions to estimate population quantities
- Interpret an estimate with its margin of error
- Compare the typical effect of random-sample size on margin of error
An estimate is not a population census
If a random sample of 400 voters gives 54% support with a margin of error of 4 percentage points, the reported interval is 50%–58%. It describes uncertainty in estimating the population proportion; it does not say that each person's opinion varies by 4% or that 54% is exact.
More data help only with the right design
Larger comparable random samples generally reduce sampling uncertainty. Under common fixed-confidence conditions, margin of error is roughly proportional to 1/√n, so quadrupling sample size roughly halves it. That relationship does not repair biased recruitment, nonresponse or misleading questions. A sample mean can estimate a population mean in the same design-based sense.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
A survey estimates 54% support with a margin of 4 percentage points. Give the interval. If a comparable sample is quadrupled, what approximate margin would you expect?
Show a starting hint
Margins stated in percentage points are added to the percentage estimate.
Compare with the explained answer
- Subtract and add 4 percentage points: 50% to 58%.
- The square root of a factor of 4 is 2.
- At the same confidence and comparable design, the margin is approximately 4/2 = 2 percentage points.
Answer / response criteria: 50%–58%; approximately 2 percentage points.
Independent practice
Independent 1
A random sample has mean waiting time 12 minutes with a margin of 1.5 minutes. Give the interval, and explain whether it contains every individual's waiting time.
Compare with the explained answer
- The interval is 12 − 1.5 to 12 + 1.5.
- This gives 10.5–13.5 minutes for the estimated population mean.
- Individual waiting times can fall outside it; it is not a range of all observations.
Answer / response criteria: 10.5–13.5 minutes; it does not bound every person's waiting time.
Application practice
Application 1
School A randomly samples 100 students and School B asks 1000 volunteers from a sports website. Which design better supports a claim about all students at School A? Would B's larger count remove its selection problem?
Compare with the explained answer
- The target is School A's student population.
- A random sample from that population supports the intended generalisation more directly.
- A large self-selected sports-website sample can remain systematically unrepresentative.
Answer / response criteria: School A's design; the larger volunteer count does not remove selection bias.
What can you explain now?
School A randomly samples 100 students and School B asks 1000 volunteers from a sports website. Which design better supports a claim about all students at School A? Would B's larger count remove its selection problem?
Compare with the explanation
School A's design; the larger volunteer count does not remove selection bias.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide