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.
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Analogy
It asks whether a poll result is far from 50 percent compared with normal poll wobble.
Example
120 of 200 prefer A, p-hat = 0.60H0: p = 0.5SE = square root of (0.25 / 200) = 0.0354z = 0.10 / 0.0354 = 2.83Two-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.