Mathfolis

Significance Test for One Proportion

Unit 6 · Inference for Categorical Data: Proportions

What AP Stats asks here

A one-proportion z-test asks whether the sample proportion is far enough from a hypothesized p0p_0 to reject the null. The trademark mistake: using p^\hat{p} in the test SE. The test imagines a world where p=p0p = p_0, so the SE under H0H_0 uses p0p_0. The CI, which makes no such assumption, uses p^\hat{p} instead.

Test setup

Hypotheses
H0:p=p0    vs.    Ha:p≠p0 (or one-sided)H_0: p = p_0 \;\;\text{vs.}\;\; H_a: p \neq p_0 \text{ (or one-sided)}
Test statistic
z=p^−p0p0(1−p0)/nz = \dfrac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}
Decision
reject H0 if p-value<α\text{reject } H_0 \text{ if p-value} < \alpha

Mnemonic

Test SE uses p0p_0
world where H0 is true\text{world where } H_0 \text{ is true}
CI SE uses p^\hat{p}
no hypothesized value\text{no hypothesized value}
AP Tip: Wording cues: 'differs' is two-sided, 'increased' or 'more than' is right-tailed, 'decreased' or 'less than' is left-tailed. Translate the claim before computing.
Caution: Never write 'accept H₀'. Failing to reject means the data are consistent with H₀ — not that H₀ is proven true.
Type 1

Set up and compute

Translate the claim into hypotheses, then compute the z-statistic with p0p_0 in the SE.

Example 1
A pharmaceutical company claims their drug helps 70% of patients. In a random sample of 150 patients, 96 improved. The test for Ha:p<0.70H_a: p < 0.70 at α=0.05\alpha = 0.05 gives: (A) z=−1.60z = -1.60, p-value ≈ 0.054, fail to reject — not enough evidence below 70%. (B) z=−1.60z = -1.60, p-value ≈ 0.054, reject because p^<0.70\hat{p} < 0.70. (C) z=−1.60z = -1.60, p-value ≈ 0.108, reject. (D) z=1.60z = 1.60, p-value ≈ 0.054, fail to reject.

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Type 2

Interpret the p-value

The p-value is the probability of data as extreme as observed assuming H0H_0 is true. It is not a probability about H0H_0 itself.

Example 2
A two-sided test gives p-value ≈ 0.0225. Which interpretation is correct? (A) There is a 2.25% probability that the true defect rate is 4%. (B) Assuming the true defect rate is 4%, the probability of observing a sample proportion as extreme as 0.06 (in either direction) is about 2.25%. (C) There is a 2.25% chance we made the wrong decision. (D) Only 2.25% of samples will give a z this large, so H0H_0 must be false.

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Type 3

Fail to reject ≠ accept

Failing to reject means the evidence is too weak to overturn H0H_0, not that H0H_0 is correct.

Example 3
A test of H0:p=0.70H_0: p = 0.70 has p-value 0.054 and the analyst fails to reject. A student concludes 'the drug really does work in 70% of patients.' Which is correct? (A) The student is correct — fail to reject means accept H0H_0. (B) Fail to reject means the data is consistent with H0H_0, not that H0H_0 is true. State 'not enough evidence to conclude p<0.70p < 0.70'; use a CI to estimate the true value. (C) The student is wrong because the p-value is too small. (D) The student is right only if α=0.10\alpha = 0.10.

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