When should the alternative hypothesis be one-sided versus two-sided?

Use a one-sided alternative (p > p_0 or p < p_0) when the problem asks about a specific direction, such as 'more than' or 'decreased,' and that direction is chosen before seeing the data. Use a two-sided alternative (p ≠ p_0) when the question asks only whether the proportion has 'changed' or is 'different.' If in doubt, two-sided is the safer default.

What is the difference between the standard error in a confidence interval and in a z-test?

A one-proportion z-test uses sqrt(p_0(1-p_0)/n) with the hypothesized proportion because the test assumes H_0 is true. A confidence interval uses p(1-p)/n with the sample proportion because there is no hypothesized value. Mixing these up is a common source of lost points.

Does a large p-value mean the null hypothesis is true?

No. Failing to reject H_0 only means the data are consistent with it — you lack convincing evidence against it. The proportion could still differ from p_0; your sample simply was not extreme enough or large enough to detect a difference. Never claim you 'accepted' or 'proved' the null hypothesis.

What must I show to earn full credit on the conditions?

State each condition by name and back it with specifics. For Random, quote the sampling or randomization described. For the 10% condition, note the sample is at most 10% of the population. For Large Counts, show the actual products np_0 and n(1-p_0) and confirm both are at least 10. Vague statements without numbers typically lose credit.

AP-STATS-6.4-6.6

U6.4 One-Proportion z-Test

Q: What is the difference between the standard error in a confidence interval and in a z-test?

A one-proportion z-test uses sqrt(p_0(1-p_0)/n) with the hypothesized proportion because the test assumes H_0 is true. A confidence interval uses p(1-p)/n with the sample proportion because there is no hypothesized value. Mixing these up is a common source of lost points.

Q: Does a large p-value mean the null hypothesis is true?

No. Failing to reject H_0 only means the data are consistent with it — you lack convincing evidence against it. The proportion could still differ from p_0; your sample simply was not extreme enough or large enough to detect a difference. Never claim you 'accepted' or 'proved' the null hypothesis.

Q: What must I show to earn full credit on the conditions?

State each condition by name and back it with specifics. For Random, quote the sampling or randomization described. For the 10% condition, note the sample is at most 10% of the population. For Large Counts, show the actual products np_0 and n(1-p_0) and confirm both are at least 10. Vague statements without numbers typically lose credit.

Master the AP Statistics one-proportion z-test: write hypotheses, check conditions, compute the z-statistic and p-value, and state a conclusion in context.

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What this lesson covers

Sometimes a claim about a population proportion sounds reasonable — a coin is fair, a candidate has majority support, a defect rate is under 5% — but sample data seem to disagree. The one-proportion z-test is the tool that decides whether that disagreement is real or just chance. This lesson walks you through the full significance-testing framework: stating hypotheses, verifying conditions, computing a test statistic, finding a p-value, and writing a conclusion that a reader could actually use. Get this structure down and you will have the template for nearly every test in the course.

Setting Up the Hypotheses

Every significance test starts with two competing claims about a population parameter — here the population proportion

p

. The null hypothesis

H_0

always states that the parameter equals a specific value, written

H_0: p = p_0

. This

p_0

is the claimed or hypothesized value; it never comes from your sample data. The alternative hypothesis

H_a

states what you suspect is true instead, and it takes one of three forms depending on the question.

Wording in the problem	Alternative
"more than", "greater than", "increased"	$H_a: p > p_0$
"less than", "fewer", "decreased"	$H_a: p < p_0$
"different", "changed", "not equal"	$H_a: p \neq p_0$

A common exam mistake is defining

p

vaguely. Always name it in context: "Let

p

= the true proportion of all registered voters in the city who support the measure." Use a two-sided alternative unless the problem clearly signals a direction before the data are collected. Never form the alternative after peeking at which way the sample leans — that inflates your error rate and loses credit on the AP exam.

Verifying the Conditions

Before computing anything, you must justify that the sampling distribution of

\hat{p}

is approximately Normal and that observations are independent. There are three conditions, and the AP rubric expects each to be stated and checked with numbers.

The Random condition requires that the data come from a random sample or randomized experiment. Quote the problem: "The 200 patients were randomly selected."

The 10% condition supports independence when sampling without replacement: the sample size must be at most 10% of the population,

n \leq 0.10N

. State it: "200 patients is less than 10% of all patients."

The Large Counts condition checks Normality. Crucially, you use the hypothesized

p_0

, not

\hat{p}

: both

np_0 \geq 10

and

n(1-p_0) \geq 10

must hold. Show the arithmetic, e.g.

200(0.15) = 30 \geq 10

and

200(0.85) = 170 \geq 10

.

Using

\hat{p}

instead of

p_0

in the Large Counts check is the single most common condition error in this topic. Remember: a test assumes

H_0

is true, so everything about the null distribution uses

p_0

Computing the z-Statistic and p-Value

Because the test assumes

H_0

, the standard deviation of

\hat{p}

uses

p_0

. The test statistic measures how many standard errors the sample proportion sits from the hypothesized value:

z = \frac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}

Here

\hat{p}

is the sample proportion,

p_0

the hypothesized proportion, and

n

the sample size. Notice the denominator uses

p_0(1-p_0)

, not

\hat{p}(1-\hat{p})

— that is what distinguishes the test's standard error from the confidence interval's.

The p-value is the probability, assuming

H_0

is true, of getting a sample result as extreme or more extreme than the one observed, in the direction of

H_a

. Find it from the standard Normal distribution. For

H_a: p > p_0

use the area to the right of

z

; for

H_a: p < p_0

the area to the left; for

H_a: p \neq p_0

double the tail area. On a calculator this is 1-PropZTest, but always report

z

and the p-value explicitly.

A larger

|z|

pushes the p-value smaller, signaling stronger evidence against

H_0

Concluding and Interpreting the p-Value

Compare the p-value to the significance level

\alpha

(often 0.05 unless stated). If p-value

\leq \alpha

, reject

H_0

; there is convincing evidence for

H_a

. If p-value

> \alpha

, fail to reject

H_0

; there is not convincing evidence for

H_a

.

Write conclusions in two linked sentences: the decision plus the evidence statement in context. For example: "Because the p-value of 0.018 is less than

\alpha = 0.05

, we reject

H_0

. We have convincing evidence that more than 15% of all patients experience the side effect."

Never say you "accept

H_0

" or that

H_0

"is true" — failing to reject only means the data are consistent with

H_0

, not that it is proven. Also avoid saying you "proved"

H_a

.

Interpret the p-value itself correctly, a frequent free-response question: it is the probability of getting a sample statistic at least as extreme as the observed value, assuming

H_0

is true. It is not the probability that

H_0

is true, and not the probability the result happened by chance in some vaguer sense. Anchor every interpretation to the phrase "assuming the null hypothesis is true."

Key terms

Null hypothesis ( $H_0$ ).: The default claim that the population proportion equals a specific value, $H_0: p = p_0$ ; the test assumes it is true when computing the p-value.
Alternative hypothesis ( $H_a$ ).: The claim the researcher seeks evidence for, stated as $p > p_0$ , $p < p_0$ , or $p \neq p_0$ .
Test statistic ( $z$ ).: The standardized distance of $\hat{p}$ from $p_0$ , measured in standard errors under the null model.
p-value.: The probability, assuming $H_0$ is true, of obtaining a sample result at least as extreme as the one observed, in the direction of $H_a$ .
Significance level ( $\alpha$ ).: A threshold chosen before analysis; reject $H_0$ when the p-value is at or below it.
Large Counts condition.: The Normality check for proportions, requiring $np_0 \geq 10$ and $n(1-p_0) \geq 10$ using the hypothesized proportion.
Standard error (test).: The estimated standard deviation of $\hat{p}$ under the null, $\sqrt{p_0(1-p_0)/n}$ , using $p_0$ rather than $\hat{p}$ .

Worked example

A nutrition company claims that 40% of adults in a region eat breakfast daily. A local health agency suspects the true proportion is lower. In a random sample of 250 adults, 88 report eating breakfast daily. Test the agency's claim at

\alpha = 0.05

Define the parameter: let

p

= the true proportion of all adults in the region who eat breakfast daily.

State hypotheses. The agency suspects the proportion is lower than 40%, so

H_0: p = 0.40

and

H_a: p < 0.40

.

Check conditions. Random: the 250 adults were randomly sampled. 10%: 250 adults is less than 10% of all adults in the region, so independence is reasonable. Large Counts:

np_0 = 250(0.40) = 100 \geq 10

and

n(1-p_0) = 250(0.60) = 150 \geq 10

. All conditions are met.

Compute

\hat{p}

\hat{p} = 88/250 = 0.352

.

Compute the test statistic:

z = \frac{0.352 - 0.40}{\sqrt{\dfrac{0.40(0.60)}{250}}} = \frac{-0.048}{\sqrt{0.00096}} = \frac{-0.048}{0.03098} \approx -1.55

Find the p-value. Since

H_a

is one-sided to the left, the p-value is the area to the left of

z = -1.55

, which is about 0.0606.

Conclude. Because the p-value of 0.0606 is greater than

\alpha = 0.05

, we fail to reject

H_0

. We do not have convincing evidence that fewer than 40% of adults in the region eat breakfast daily.

Practice questions

A researcher tests

H_0: p = 0.5

against

H_a: p \neq 0.5

and obtains a test statistic of

z = 2.10

. Which of the following is the correct p-value?

0.0179
0.0357
0.9821
0.4821

Answer: 0.0357

The alternative is two-sided, so the p-value is twice the area beyond

z = 2.10

. The area to the right of 2.10 is about 0.0179, so the p-value is

2(0.0179) = 0.0357

. Choosing 0.0179 forgets to double for the two tails; 0.9821 is the area to the left, which is not a p-value.

When checking the Large Counts condition for a one-proportion z-test, why do we use

p_0

instead of

\hat{p}

Answer: Because a significance test evaluates the sampling distribution assuming the null hypothesis is true, so every quantity describing that null distribution — including the Normality check — uses the hypothesized value $p_0$ .

The whole logic of the test is conditional on

H_0

being true. The sampling distribution of

\hat{p}

under the null is centered at

p_0

with standard deviation

\sqrt{p_0(1-p_0)/n}

, so the Large Counts check must also use

p_0

. This mirrors why the standard error in the z-statistic uses

p_0(1-p_0)

, not

\hat{p}(1-\hat{p})

. Confidence intervals differ because they have no hypothesized value and instead use

\hat{p}

A student writes: "The p-value is 0.03, so there is a 3% chance the null hypothesis is true." Explain what is wrong with this interpretation and give a correct one.

Answer: The p-value is not the probability that $H_0$ is true. Correctly: if $H_0$ were true, there is a 0.03 probability of obtaining a sample proportion at least as extreme as the one observed.

A p-value is a conditional probability computed assuming

H_0

is true; it describes how surprising the data are under that assumption, not the probability the hypothesis itself is true. Hypotheses are statements about a fixed population parameter, which is either true or false — it does not have a probability in this framework. The correct interpretation always begins with 'assuming the null hypothesis is true.'

FAQ

When should the alternative hypothesis be one-sided versus two-sided?: Use a one-sided alternative ( $p > p_0$ or $p < p_0$ ) when the problem asks about a specific direction, such as 'more than' or 'decreased,' and that direction is chosen before seeing the data. Use a two-sided alternative ( $p \neq p_0$ ) when the question asks only whether the proportion has 'changed' or is 'different.' If in doubt, two-sided is the safer default.
What is the difference between the standard error in a confidence interval and in a z-test?: A one-proportion z-test uses $\sqrt{p_0(1-p_0)/n}$ with the hypothesized proportion because the test assumes $H_0$ is true. A confidence interval uses $\sqrt{\hat{p}(1-\hat{p})/n}$ with the sample proportion because there is no hypothesized value. Mixing these up is a common source of lost points.
Does a large p-value mean the null hypothesis is true?: No. Failing to reject $H_0$ only means the data are consistent with it — you lack convincing evidence against it. The proportion could still differ from $p_0$ ; your sample simply was not extreme enough or large enough to detect a difference. Never claim you 'accepted' or 'proved' the null hypothesis.
What must I show to earn full credit on the conditions?: State each condition by name and back it with specifics. For Random, quote the sampling or randomization described. For the 10% condition, note the sample is at most 10% of the population. For Large Counts, show the actual products $np_0$ and $n(1-p_0)$ and confirm both are at least 10. Vague statements without numbers typically lose credit.

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