The 1-prop z-test (one-sample z-test for a population proportion) is the AP Stats significance test that uses z = (p̂ − p₀)/√(p₀(1−p₀)/n) to decide whether sample data give convincing evidence that the true population proportion p differs from a hypothesized value p₀.
A 1-prop z-test answers one question. Someone claims the true proportion of a population is some value p₀ (say, "60% of students recycle"). You take a random sample, get a sample proportion p̂, and ask whether your p̂ is far enough from p₀ that random chance alone is a bad explanation. The test standardizes the gap with a z-score, z = (p̂ − p₀) / √(p₀(1−p₀)/n), then converts that z into a P-value using the standard normal distribution.
Notice the standard error uses p₀, not p̂. That's because a hypothesis test starts by assuming the null hypothesis is true, so you build the sampling distribution around the claimed value. Per the CED (AP Stats 6.4.B), this is the appropriate testing method whenever you have one categorical variable and one sample. Before you compute anything, you have to verify conditions (AP Stats 6.4.C): the data come from a random sample, n ≤ 10% of the population if sampling without replacement, and both np₀ ≥ 10 and n(1−p₀) ≥ 10 so the sampling distribution of p̂ is approximately normal.
This is the centerpiece of Unit 6 (Inference for Categorical Data: Proportions) and the direct payoff of Topic 6.4, Setting Up a Test for a Population Proportion. It pulls together three learning objectives at once. AP Stats 6.4.A has you write H₀: p = p₀ against a one-sided (< or >) or two-sided (≠) alternative based on the question. AP Stats 6.4.B has you name the procedure, and the name matters since "one-sample z-test for a population proportion" is exactly the language graders look for. AP Stats 6.4.C has you check independence and normality conditions before trusting the math. It's also your first full significance test in the course, so the four-step structure you learn here (hypotheses, conditions, mechanics, conclusion) is the template for every test that follows in Units 6-9.
Keep studying AP® Statistics Unit 3
Null and Alternative Hypotheses (Unit 6)
Every 1-prop z-test starts with H₀: p = p₀ and an alternative built from the question of interest. The direction of Hₐ (one-sided vs. two-sided) decides whether your P-value is one tail or both tails, so getting the hypotheses right changes your final answer.
Large Counts Condition (Units 5-6)
Here's the twist from confidence intervals. For the test you check np₀ ≥ 10 and n(1−p₀) ≥ 10 using the hypothesized p₀, because the whole test assumes H₀ is true. The interval version checks the same idea with p̂ instead.
Z-Score (Units 1, 5, 6)
The test statistic is literally a z-score for p̂ on its sampling distribution. The same "how many standard deviations from center" logic from Unit 1 powers the whole test; Unit 5 told you that distribution is approximately normal, which is why a z works.
P-Value (Unit 6)
The z-statistic is just the middle step. The P-value is the probability of getting a p̂ at least as extreme as yours if p really equals p₀, and comparing it to α (usually 0.05) is what actually drives your reject-or-fail-to-reject conclusion.
On multiple choice, expect stems that test the setup more than the arithmetic. You'll identify the correct hypotheses, spot a violated condition, pick the right standard error formula (the one with p₀ in it), or interpret a P-value in context. On the free response, the proportion significance test is a classic full-inference question, and you're graded on all four steps. State H₀ and Hₐ with p defined in context, name the procedure ("one-sample z-test for a population proportion") and verify random sampling, the 10% condition, and large counts with p₀, compute z and the P-value, then write a conclusion that compares the P-value to α and answers the question in context. Calculator-wise, the 1-PropZTest function on a TI handles the mechanics and returns z and the P-value, but the calculator never checks conditions or writes conclusions. Those points are all on you.
Both involve one sample and one proportion, but they answer different questions with slightly different math. The test assumes a claimed value p₀ and asks "is my data inconsistent with that claim?", so its standard error uses p₀: √(p₀(1−p₀)/n). The interval makes no claim and estimates p, so its standard error uses p̂: √(p̂(1−p̂)/n). Same logic for conditions, where the test checks large counts with p₀ and the interval checks them with p̂. If the problem says "test the claim" or "is there convincing evidence," run the test; if it says "estimate" or "construct an interval," build the interval.
A 1-prop z-test checks whether a sample proportion p̂ provides convincing evidence that the true population proportion differs from a hypothesized value p₀.
The null hypothesis is always H₀: p = p₀, and the alternative uses <, >, or ≠ depending on what the question is actually asking.
The standard error in the test statistic uses p₀, not p̂, because the test is built on the assumption that the null hypothesis is true.
Before running the test, verify three things: the data come from a random sample or randomized experiment, n is at most 10% of the population, and np₀ and n(1−p₀) are both at least 10.
The CED name for this procedure is the one-sample z-test for a population proportion, and naming it correctly earns credit on FRQs.
The calculator's 1-PropZTest gives you z and the P-value, but hypotheses, condition checks, and an in-context conclusion are graded separately and must be written out.
It's the significance test for a single population proportion. You assume the true proportion equals a claimed value p₀, compute z = (p̂ − p₀)/√(p₀(1−p₀)/n), and use the resulting P-value to decide whether your sample gives convincing evidence against that claim. It lives in Topic 6.4 of Unit 6.
Use p₀. A significance test assumes H₀ is true, so the sampling distribution is centered at p₀ with standard error √(p₀(1−p₀)/n). Using p̂ in the standard error is the interval formula, not the test formula, and mixing them up costs points.
The test evaluates a specific claim about p (does the data contradict p = p₀?), while the interval estimates p with no claim involved. The test uses p₀ in its standard error and large counts check; the interval uses p̂ for both.
With proportions, the standard deviation of the sampling distribution depends only on p₀ and n, both of which are known once you assume H₀. There's no unknown σ to estimate, which is what forces a t-distribution for means in Unit 7. Proportions always use z.
No. The calculator only handles the mechanics, returning z and the P-value. On the FRQ you also need stated hypotheses with p defined in context, verified conditions (random, 10%, large counts with p₀), and a conclusion linking the P-value to α and the original question. Most of the rubric points come from those parts.
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