Unit 6: Inference for Categorical Data: Proportions

Confidence Interval for a Proportion

Constructing and interpreting a one-sample confidence interval for a proportion.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Estimating an Unknown Proportion, with a Margin of Error

A random sample of 200 voters finds 112 support a proposition. The true proportion of all voters who support it is unknown — but the sampling distribution from the last unit describes exactly how much p̂ should vary around that unknown truth. Before reading on: could that same variability be used to build a range of plausible values for the true proportion, instead of just a single guess?

Definition — Confidence Interval for a Proportion

A confidence interval for p has the form p̂ ± z*·√(p̂(1−p̂)/n) — a point estimate plus or minus a margin of error. Conditions: the sample must be random, the sample size no more than 10% of the population (so sampling without replacement doesn't distort independence), and np̂ ≥ 10 and n(1−p̂) ≥ 10 (large counts, using p̂ since the true p is unknown here).

\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Worked Example — Constructing and Interpreting a 95% Confidence Interval

p̂ = 112/200 = 0.56. Conditions: random sample (given), 10% condition (assume a large voter population), large counts: 200(0.56)=112≥10 and 200(0.44)=88≥10 — all satisfied. SE = √(0.56×0.44/200) ≈ 0.0351. For 95% confidence, z* = 1.96. Margin of error = 1.96 × 0.0351 ≈ 0.0688. The interval: 0.56 ± 0.0688 = (0.4912, 0.6288), or about (49.1%, 62.9%).

The correct interpretation: 'We are 95% confident that the true proportion of all voters who support the proposition is between 49.1% and 62.9%.' This means: 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 in this specific interval — the true proportion is a fixed (if unknown) number; it's either in this particular interval or it isn't. The 95% describes the reliability of the method used to produce the interval, not a probability about this one result.

Tip

A higher confidence level (say, 99% instead of 95%) widens the interval — capturing the true parameter more reliably requires a bigger net. A larger sample size narrows the interval instead, since SE shrinks as n grows.

Common Mistakes

Interpreting a 95% confidence interval as 'there's a 95% probability the true proportion is between 49.1% and 62.9%.'
The true proportion is fixed, not random — the 95% describes how often the sampling-and-interval-construction procedure succeeds across repeated samples, not a probability about this one already-computed interval.
Using the population proportion's true value (if it happens to be known from elsewhere) instead of p̂ in the standard error formula.
A confidence interval is built specifically because the true p is unknown — its standard error formula uses p̂, the best available estimate, not a hypothesized or externally known value.

Key Takeaways

A confidence interval for a proportion is p̂ ± z*·SE, requiring random sampling, the 10% condition, and large counts.
The correct interpretation refers to the reliability of the repeated-sampling procedure, never a probability statement about the fixed true parameter.
Higher confidence widens an interval; larger sample size narrows it.

Summary

A confidence interval estimates a plausible range for an unknown proportion. The next lesson uses the same sampling distribution machinery to test a specific claimed value instead.

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