Nova Academy — Foundations of Inquiry | Lesson 020 | November 9, 2026
Learning objectives
- Explain what a confidence interval estimates and what its confidence level means.
- Compute an approximate interval for a sample proportion under stated assumptions.
- Distinguish sampling uncertainty from selection bias and measurement error.
From variation to an interval
Lesson 019 showed why repeated random samples from the same population disagree. Rather than pretending one sample result is exact, we can report a range produced by a statistical procedure. A 95% confidence interval is constructed by a method that, under its assumptions, captures the fixed population parameter in about 95% of repeated samples. It does not mean there is a 95% probability that the fixed parameter lies in this particular calculated interval.
Worked example: a proportion
Suppose a simple random sample of 400 eligible respondents includes 240 who prefer option A. The observed proportion is 240/400 = 0.60. For an introductory normal-approximation (Wald) interval, the estimated standard error is √[0.60 × 0.40 / 400] = √0.0006 ≈ 0.0245. The approximate 95% margin of error is 1.96 × 0.0245 ≈ 0.0480. The interval is 0.60 ± 0.0480, or approximately 55.2% to 64.8%.
This simple method works reasonably for this illustrative large, independent random sample, but the Wald interval can perform poorly for small samples or proportions near zero or one. For practical reporting, Wilson or exact binomial methods are often preferable. Nonresponse, selection bias, and bad questions can undermine conclusions even when the arithmetic is correct.
Activity: change the sample size
Keep the observed proportion at 0.60, but imagine 1,600 independent random responses instead of 400. Calculate the standard error and approximate 95% interval. What changes? What does not?
Knowledge check
- Does 95% confidence mean 95% of individuals fall inside the interval?
- If the sample size quadruples, approximately what happens to the standard error, with the sample proportion held constant?
- Can a narrow confidence interval repair a biased sampling frame?
Answer key
Activity: √(0.60 × 0.40 / 1600) = √0.00015 ≈ 0.01225. Margin ≈ 1.96 × 0.01225 = 0.0240. Interval ≈ 57.6% to 62.4%. The margin is about half as wide, but the underlying risks of bias and measurement error remain. Knowledge check: (1) No: the interval concerns a population parameter, not individual outcomes. (2) It halves. (3) No.
Reflection
When you hear “the margin of error is three points,” what questions should you ask about who was sampled, how they were reached, and what they were asked?
Sources and further reading
- NIST/SEMATECH e-Handbook of Statistical Methods — sampling, uncertainty, and interval estimation.
- American Statistical Association: confidence intervals and interpretation — supplemental conceptual guidance.
- OpenStax Introductory Statistics 2e — freely available textbook treatment of confidence intervals.
