Setting up a one-sample $z$ test for a population proportion means writing the null and alternative hypotheses, naming the right test, and checking the conditions before you calculate anything. The null is always $H_0: p = p_0$ , and your alternative uses $<$ , $>$ , or $\ne$ depending on what the question is looking for.

Why This Matters for the AP Statistics Exam

Unit 6 covers about 12 to 15 percent of the exam, and proportion tests show up in both multiple-choice and free-response work. Topic 6.4 is the setup stage, so getting it right sets you up for the later steps of finding the p-value and stating a conclusion.

On the exam you will need to:

Recognize when a one-sample z-test for a proportion is the correct procedure for a single categorical variable.
Write clear hypotheses using correct notation.
Verify conditions and show that work, which is important for clear, full-credit free-response answers.

This topic does not ask you to finish the test. It asks you to get the foundation right so the rest of the test holds up.

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Key Takeaways

The null hypothesis for a proportion is always H₀: p = p₀, where p₀ is the claimed value.
The alternative hypothesis uses <, >, or ≠ based on the question. One-sided uses < or >; two-sided uses ≠.
A one-sided null may include ≥ or ≤, but it is still tested at the boundary of equality (p = p₀).
The correct procedure for one categorical variable is a one-sample z-test for a population proportion.
Check two condition groups: independence (random data plus the 10% condition) and approximate normality (the large counts check).
For the normality check in a test, use p₀, not p̂: confirm np₀ ≥ 10 and n(1 − p₀) ≥ 10.

Writing Your Hypotheses

Every one-sample z-test for a proportion starts with two hypotheses: the null and the alternative.

Null Hypothesis (H₀)

The null hypothesis is the statement assumed to be true unless the data give strong evidence against it. It usually represents no difference or no effect. For a proportion, it is always written as:

H₀: p = p₀

Here p₀ is the specific value claimed in the problem. For example, if a claim says the population proportion is 0.5, then H₀: p = 0.5. You assume that value is correct until the sample data suggest otherwise.

Alternative Hypothesis (Ha)

The alternative hypothesis is the statement you are collecting evidence for. It always uses a strict inequality:

Ha: p < p₀ (one-sided, lower)
Ha: p > p₀ (one-sided, upper)
Ha: p ≠ p₀ (two-sided)

Use < or > when the question asks whether the true proportion is below or above the claimed value. Use ≠ when the question only asks whether the true proportion is different from the claimed value.

A quick note on direction: the null hypothesis for a one-sided test may be written with ≥ or ≤, but it is still tested at the boundary of equality, meaning p = p₀.

Example

An article states that 94% of all people can identify the pop culture icon Baby Yoda. To test this claim, you poll a random sample of 750 people and find that 700 of them can correctly identify Baby Yoda. Do the data give evidence that the actual proportion is less than 94%?

This is a population claim tested with a sample, so the right procedure is a one-sample z-test for a population proportion. Write the hypotheses:

H₀: p = 0.94
Ha: p < 0.94

Checking Conditions

Before you run any calculations, you have to verify the conditions. Skipping this step or stating it vaguely costs you on free-response questions.

Independence

Your data must come from a random sample or a randomized experiment. Without randomness, your sample could be biased, and no calculation fixes bias.

When sampling without replacement, also check the 10% condition: the sample size should be no more than 10% of the population (n ≤ 0.10N). This keeps the selections close enough to independent. State it like this: "It is reasonable to believe the population is at least 10 times the sample size, so independence is satisfied."

Approximate Normality

You are using the normal curve to model the sampling distribution of p̂, so you need it to be approximately normal. For a test, check the large counts condition using p₀ (the null value), not the sample proportion:

np₀ ≥ 10
n(1 − p₀) ≥ 10

Both expected successes and expected failures must be at least 10.

Example

For the Baby Yoda problem:

Independence: "We poll a random sample of 750 people," and it is reasonable that the population is at least 7,500 people, so the 10% condition holds.
Approximate normality: 750(0.94) = 705 expected successes and 750(0.06) = 45 expected failures. Both are at least 10, so the condition is met.

Looking Ahead: Test Statistic and p-Value

Topic 6.4 stops at setup, but it helps to see where this leads. Once your hypotheses and conditions are set, the next step (Topic 6.5) is calculating the test statistic and p-value.

The standardized test statistic for a proportion is:

z = (p̂ − p₀) / √(p₀(1 − p₀)/n)

Notice that the standard error in a test uses p₀, the null value, not p̂. That matches the normality check, where you also used p₀. After you find z, you use the standard normal distribution to find the p-value, then compare it to your significance level to make a decision.

Your graphing calculator can speed this up. In the Stats Tests menu, choose 1-Prop Z-Test, enter your values, and it returns the z-score and p-value. Even when you use technology, still write your hypotheses and conditions clearly so your reasoning is visible.

How to Use This on the AP Statistics Exam

Free Response

Define your parameter in context. State what p represents, such as "p = the true proportion of all people who can identify Baby Yoda."
Write both hypotheses with correct notation and the right p₀ value.
Match the inequality in Ha to the question. "More than" means >, "less than" means <, and "different from" means ≠.
Show both condition checks with actual numbers, not just labels. Write out np₀ and n(1 − p₀) and compare them to 10.

MCQ

Be ready to pick the correct hypotheses from a list. Watch for the direction of the inequality.
Know that the normality check for a test uses p₀, not p̂.
Recognize a one-sample z-test for a proportion as the right method for a single categorical variable.

Common Trap

Putting an inequality in the null. The null for a proportion is always written as an equality at the boundary, H₀: p = p₀.

Common Misconceptions

Using p̂ in the normality check for a test. In a test you assume H₀ is true, so use p₀: np₀ ≥ 10 and n(1 − p₀) ≥ 10. The sample proportion p̂ is used for confidence intervals, not for the test condition.
Choosing the alternative direction from the sample data. The direction of Ha comes from the research question, not from whether p̂ happened to land above or below p₀.
Thinking the 10% condition is about the sample being large. It is about keeping selections approximately independent when sampling without replacement, so n must be small relative to the population.
Believing a one-sided test means the null has an inequality you actually test. Even when the null is written with ≥ or ≤, it is tested at the boundary of equality, p = p₀.
Skipping the parameter definition or context. Hypotheses without a clear definition of p in context are incomplete on free-response questions.

Vocabulary

The following words are mentioned explicitly in the College Board Course and Exam Description for this topic.

Term	Definition
10% condition	The requirement that sample size n is at most 10% of the population size N to ensure independence when sampling without replacement.
alternative hypothesis	The claim that contradicts the null hypothesis, representing what the researcher is trying to find evidence for.
approximately normal	A distribution that closely follows the shape of a normal distribution, allowing for the use of normal probability methods.
categorical variable	A variable that takes on values that are category names or group labels rather than numerical values.
independence	The condition that observations in a sample are not influenced by each other, typically ensured through random sampling or randomized experiments.
null hypothesis	The initial claim or assumption being tested in a hypothesis test, typically stating that there is no effect or no difference.
number of failures	The count of unfavorable outcomes in a sample, denoted as n(1-p̂), used to verify the normality condition.
number of successes	The count of favorable outcomes in a sample, denoted as np̂, used to verify the normality condition.
one-sample z-test for a population proportion	A hypothesis test used to determine whether a sample proportion provides evidence that a population proportion differs from a hypothesized value.
one-sided alternative hypothesis	An alternative hypothesis that specifies the direction of the difference, either p₁ < p₂ or p₁ > p₂.
population proportion	The true proportion or percentage of a characteristic in an entire population, typically denoted as p.
random sample	A sample selected from a population in such a way that every member has an equal chance of being chosen, reducing bias and allowing for valid statistical inference.
randomized experiment	A study design where subjects are randomly assigned to treatment groups to establish cause-and-effect relationships.
sample proportion	The proportion of individuals in a sample that have a particular characteristic, denoted as p-hat (p̂).
sampling distribution	The probability distribution of a sample statistic (such as a sample proportion) obtained from repeated sampling of a population.
sampling without replacement	A sampling method in which an item selected from a population cannot be selected again in subsequent draws.
statistical inference	The process of drawing conclusions about a population based on data collected from a sample.
two-sided alternative hypothesis	An alternative hypothesis that specifies the difference could be in either direction, stated as p₁ ≠ p₂.

Frequently Asked Questions

How do you set up a test for a population proportion?

Define the parameter p in context, write H0: p = p0, choose the correct alternative hypothesis, identify the procedure as a one-sample z-test for a population proportion, and check the required conditions.

What is H0 for a population proportion test?

The null hypothesis is H0: p = p0, where p0 is the claimed population proportion. The null is tested at the boundary of equality.

How do you choose Ha for a population proportion test?

Use Ha: p < p0 for a lower one-sided test, Ha: p > p0 for an upper one-sided test, and Ha: p != p0 when the question asks whether the proportion is different.

When do you use a one-sample z-test for a population proportion?

Use a one-sample z-test for a population proportion when you have one categorical variable and want to test a claim about a single population proportion.

What conditions do you check for a population proportion test?

Check independence using random data and, when sampling without replacement, the 10% condition. Then check large counts using the null value: np0 >= 10 and n(1 - p0) >= 10.

How is AP Stats 6.4 tested?

AP Stats 6.4 is tested through hypothesis notation, procedure selection, and condition checks. Free-response answers should define p in context and show the random, 10%, and large counts checks with numbers.