# Test for a proportion

A one-proportion z test checks a claimed share p0. The statistic is z = (p-hat - p0) / square root of (p0(1 - p0) / n). Use the claimed p0, not p-hat, in the standard error.

Statistics (university) · Part 06: Hypothesis testing · Card 9 of 9 · Last updated 2026-10-19

## Analogy

It asks whether a poll result is far from 50 percent compared with normal poll wobble.

## Example

**Is it really 50 percent?**

120 of 200 prefer A, p-hat = 0.60
H0: p = 0.5
SE = square root of (0.25 / 200) = 0.0354
z = 0.10 / 0.0354 = 2.83
Two-sided p = 0.0047

## In practice

Tests whether a defect rate has risen, or whether more than half of customers prefer a product.

## Practice

### Flashcards

- **Which proportion goes in the standard error of a z test?** The claimed proportion p0.

## Questions people ask

### Why use p0 and not p-hat in the standard error?

A test assumes the null is true, so we measure the wobble that would occur if the share really were p0. Using p0 keeps the calculation consistent with that assumption. Confidence intervals have no null, so they use p-hat instead.

### What conditions are needed for the z test for a proportion?

Observations should be independent, from a random sample. Both n times p0 and n times (1 - p0) should be at least 10, so the normal curve is a good fit. With fewer, use an exact binomial test, which calculates the probability directly instead of relying on the normal curve.

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