Significance Tests for a Proportion

You’ll be able to

State null and alternative hypotheses for a proportion
Compute the one-sample z test statistic and p-value
Make and interpret a decision in context

Hypotheses

A significance test weighs evidence against a claim. The null hypothesis H₀ states "no effect / the status quo," always as an equality: H₀: p = p₀. The alternative Hₐ states what we suspect: p < p₀, p > p₀, or p ≠ p₀. Hypotheses are always about the parameter p, never the statistic p-hat. Set them before looking at the data.

Test statistic and p-value

The z test statistic measures how many standard deviations p-hat falls from the null value p₀, using the null proportion in the standard error (because we assume H₀ true). The p-value is the probability, assuming H₀ is true, of getting a result as extreme or more extreme than observed. A small p-value means the data would be surprising if H₀ were true — evidence against H₀.

One-sample z test for a proportion

z = (p-hat − p₀) / sqrt( p₀(1−p₀) / n )

Use the null value p₀ (not p-hat) in the standard error, since the p-value is computed assuming H₀ is true. Check Random, 10%, and Large Counts (np₀ ≥ 10, n(1−p₀) ≥ 10).

Worked example

A company claims 90% of orders ship on time. An SRS of 200 orders finds 168 on time (p-hat = 0.84). Test at α = 0.05 whether the true on-time rate is below 90%.

1.Hypotheses: H₀: p = 0.90 vs. Hₐ: p < 0.90. Conditions: random, 10% (population > 2000), np₀ = 180 and n(1−p₀) = 20 both ≥ 10.
2.Standard error (using p₀): sqrt(0.90·0.10/200) = sqrt(0.00045) ≈ 0.0212.
3.Test statistic: z = (0.84 − 0.90)/0.0212 ≈ −2.83.
4.p-value: P(Z < −2.83) ≈ 0.0023, which is less than α = 0.05.

Answer: z ≈ −2.83, p-value ≈ 0.0023 < 0.05, so we reject H₀. There is convincing evidence that the true proportion of on-time orders is less than 0.90.

Checkpoint

A significance test yields a p-value of 0.03. Using α = 0.05, what is the correct conclusion?

Watch out

The p-value is not the probability that H₀ is true. It is the probability of data as extreme as observed assuming H₀ is true. And we never "accept" H₀ — we either "reject H₀" or "fail to reject H₀," because absence of evidence is not proof of no effect.

Checkpoint

In the standard error for a one-sample z test for a proportion, which value should be used inside the square root?

Answer the 2 checkpoints as you read.