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Introduction to Inference

Confidence Intervals

Constructing and interpreting a confidence interval for a population proportion.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Sampling Distributions

Estimating an Unknown Proportion from One Sample

A random sample of 80 students at a school finds 52 prefer online classes over in-person. Before reading on: since the true proportion of all students who prefer online classes is unknown, could the previous lesson's sampling-distribution ideas be used to build a reasonable range of plausible values for it, instead of just reporting this one sample's result?

Definition — Confidence Interval for a Proportion

A confidence interval has the form p̂ ± z*·√(p̂(1−p̂)/n) — a sample estimate plus or minus a margin of error built from the sampling distribution's typical spread. Conditions: the sample should be random, and both n·p̂ and n·(1−p̂) should be at least 10.
p^±z∗p^(1−p^)n\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Worked Example — Constructing and Interpreting a 95% Confidence Interval

p̂ = 52/80 = 0.65. Conditions: random sample (given); 80(0.65) = 52 ≥ 10 and 80(0.35) = 28 ≥ 10 — satisfied. SE = √(0.65×0.35/80) ≈ 0.0533. For 95% confidence, z* = 1.96. Margin of error = 1.96 × 0.0533 ≈ 0.105. The interval: 0.65 ± 0.105 = (0.545, 0.755), or about (54.5%, 75.5%).

The correct interpretation: 'We are 95% confident that between about 54.5% and 75.5% of all students at the school prefer online classes.' This means that if this same random sampling procedure were repeated many times, each producing its own confidence interval, about 95% of those intervals would capture the true population proportion. It does NOT mean there's a 95% probability the true proportion falls specifically within this one interval — the true proportion is a fixed, if unknown, number; the 95% describes how reliable the method is across repeated use, not a probability about this single already-computed result.

Tip

A wider confidence interval isn't a worse result — it's the honest cost of either a higher confidence level or a smaller sample size, both of which genuinely reduce how precisely the true proportion can be pinned down.

Common Mistakes

  • Interpreting the interval as 'there's a 95% probability the true proportion is between 54.5% and 75.5%.'

    The true proportion is fixed, not random — the 95% describes the reliability of the sampling-and-interval procedure across repeated use, not a probability statement about this one already-computed interval.

Key Takeaways

  • A confidence interval for a proportion combines a point estimate with a margin of error built from the sampling distribution's spread.
  • The correct interpretation refers to how often the procedure succeeds across repeated sampling — never a probability about the one fixed true value.
  • Confidence intervals give a plausible range for an unknown population value, acknowledging the uncertainty a single sample can't eliminate.

Summary

This closes the Statistics course: exploring one- and two-variable data, collecting it thoughtfully, reasoning about probability and random variables, and using sampling distributions to estimate an unknown population proportion — each building on the last toward genuine statistical reasoning about data and uncertainty.

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