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
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide