AP Statistics Topic 3.6: p-Values

01 · Ask how unusual the result would be

The concept in 60 seconds

A school investigates whether more than $60\%$ of its students prefer an earlier lunch. An SRS of $500$ students from $10{,}000$ students, taken without replacement, gives $320$ yes responses. The observed sample proportion is $\hat{p} =\frac{320}{500}= 0.64$ .

Let $p$ be the proportion of all students at this school who prefer an earlier lunch. The hypotheses are $H_{0}: p = 0.60$ and $H_{a}: p > 0.60$ . If the population really had $60\%$ support, different random samples would still give different proportions. The test asks how often the null model would give a result at least as high as $0.64$ .

Visual guide 1: the

p

-value question in three steps

Assume $H_{0}: p = 0.60$

Build the school’s null model at $60\%$ preference.

CompareObserved $\hat{p} = 0.64$

Ask about this high result and results even higher.

Measure $\text{Right-tail area} \approx 0.0339$

For $H_{a}: p > 0.60$ , measure the null probability to the right.

The probability concerns possible sample results under $H_{0}$ . It is not the probability that $H_{0}$ itself is true.

A $p$ -value is the probability, assuming $H_{0}$ is true, of a test statistic at least as extreme as the observed statistic in the direction specified by $H_{a}$ .

For this school’s one-sided normal $z$ -test, the $p$ -value is approximately $0.0339$ , or $3.39\%$ . Assuming the true preference proportion is $0.60$ , about $3.39\%$ of random samples of this size would produce a result this high or higher, according to the approximate null model.

That is a relatively unusual result under the $60\%$ model, providing evidence for a larger population proportion. It does not tell us the probability that the $60\%$ claim is true.

All contexts are fictional teaching examples. Normal tail probabilities are approximations for the one-proportion $z$ -test. The simulation later uses generated null-model statistics, not collected school responses.

Quick check: what do we assume when calculating the school’s

p

-value?

Assume $H_{0}: p = 0.60$ is true, together with the justified sampling model. We evaluate sample results under that assumption; we do not calculate the probability that the assumption itself is true.

02 · Keep the assumption and the event separate

What is the probability about?

The phrase “assuming $H_{0}$ is true” is essential. It tells us which model supplies the probability. The event is a sample test statistic in the extreme region determined by $H_{a}$ .

\text{p-value}=P(\text{statistic at least as extreme as observed}\mid H_0\text{ and the model assumptions})

The bar $\mid$ means “given” or “assuming.”

Read this in order: first assume the null model; then find the probability of the specified sample results. Reversing that order changes the question. A probability about data under a hypothesis is different from a probability about the hypothesis after observing data.

Visual guide 2: similar names, different roles

Population parameter $p$ is unknown

The actual whole-school preference proportion.

Null benchmark $p_{0} = 0.60$

The fixed value used to build the null model.

Conditional probability $\text{p-value} \approx 0.0339$

A tail probability under that model, for this observed result and alternative.

The sample statistic $\hat{p} = 0.64$ is the observed evidence. It differs from the population parameter, the null benchmark and the $p$ -value.

Read each symbol by its role in the school example.
Symbol or term	Meaning	School example
$p$	Unknown population proportion	Proportion of all school students preferring earlier lunch
$p_{0}$	Null population benchmark	$0.60$
$\hat{p} =\frac{x}{n}$	Observed sample proportion	$\frac{320}{500}= 0.64$
$z_{\mathrm{obs}}$	Observed standardized test statistic	Approximately $+1.8257$
$Z$	Random statistic under the null approximation	Standard normal: center $0$ , SD $1$
$p$ -value	Null probability of the specified extreme region	$\text{Right-tail probability} \approx 0.0339$
$B$	Number of simulated null statistics	$10{,}000$ in the illustrated run

Three different uses of “ $p$ ”

In this unit, $p$ alone usually names the unknown population proportion. $p_{0}$ is the numerical proportion specified by $H_{0}$ . The $p$ -value is a calculated probability about the test statistic. The words “ $p$ -value” keep its role clear.

In the school example, $p$ is unknown, $p_{0} = 0.60$ , $\hat{p} = 0.64$ and the right-tail $p$ -value is approximately $0.0339$ . These $4$ quantities answer different questions.

Do not reverse the condition: “Assuming $60\%$ of students prefer earlier lunch, how unusual is this sample?” is the question the $p$ -value answers. “Given this sample, what is the chance that exactly $60\%$ prefer earlier lunch?” needs a different framework and is not answered by this $p$ -value.

Quick check: does

\text{p-value} = 0.0339

mean there is a

3.39\%

chance that

p = 0.60

No. The calculation assumes $p = 0.60$ . The $3.39\%$ is the approximate probability of a sample result as high as the observed result or higher under that assumption.

03 · Build the reference before judging the result

The null model and extreme results

A null distribution describes how the test statistic would vary across repeated samples if $H_{0}$ were true. For a justified one-proportion $z$ -test, the standardized statistic is approximately standard normal under $H_{0}$ : centered at $0$ with standard deviation $1$ .

The school result corresponds to a supplied observed statistic of $z \approx +1.8257$ . A positive $z$ means the observed proportion is above the null benchmark. A negative $z$ means it is below. Calculating $z$ from the sample is developed in Topic 3.7; here we use it to select and interpret probability areas.

What makes a result “as extreme or more extreme”?

For $H_{a}: p > p_{0}$ : results at least as far toward larger proportions count. Use $Z \ge z_{\mathrm{obs}}$ .
For $H_{a}: p < p_{0}$ : results at least as far toward smaller proportions count. Use $Z \le z_{\mathrm{obs}}$ .
For $H_{a}: p \ne p_{0}$ : departures at least as large in either direction count. For this symmetric normal model, use $\lvert Z\rvert \ge \lvert z_{\mathrm{obs}}\rvert$ .

The $p$ -value includes a whole region of outcomes. It is not the probability of exactly the observed value. On a continuous normal model, $1$ point has probability $0$ ; a tail area can have positive probability.

Check that the reference model is appropriate

The school uses an SRS, satisfies $500 \le 0.10(10{,}000) = 1{,}000$ , and has null expected counts $500(0.60) = 300$ and $500(0.40) = 200$ , both at least $10$ . These support the usual normal approximation.

A calculation cannot repair a biased sampling design or a poor probability model. If the normal approximation is unsuitable, a suitable exact or null simulation method may give a different probability. Do not treat a normal $z$ -test’s $p$ -value as an exact binomial probability.

Quick check: does the right-tail school

p

-value include results above

0.64

Yes. It includes $0.64$ and higher in the approximate sampling model. “At least as extreme” includes more extreme results, not just the sample result we happened to observe.

04 · Read

H_{a}

before choosing an area

Choose the correct tail area

Let $Z$ follow the standard normal null approximation, and let $z_{\mathrm{obs}}$ be the observed statistic. The sign in $H_{a}$ tells you which probability to calculate.

Use the alternative to choose the null probability region.
Alternative	Extreme region for $Z$	Normal-model calculation
$H_{a}: p > p_{0}$	$Z \ge z_{\mathrm{obs}}$	$\text{Right area}: 1 – \Phi (z_{\mathrm{obs}})$
$H_{a}: p < p_{0}$	$Z \le z_{\mathrm{obs}}$	$\text{Left area}: \Phi (z_{\mathrm{obs}})$
$H_{a}: p \ne p_{0}$	$Z\le-\lvert z_{\mathrm{obs}}\rvert\text{ or }Z\ge\lvert z_{\mathrm{obs}}\rvert$	$\text{Both outer tails}: 2[1 – \Phi (\lvert z_{\mathrm{obs}}\rvert )]$

Visual guide 3:

1

observed result,

3

possible questions

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$p$ -Values

By the end of this lesson, you should be able to:

The concept in 60 seconds

What is the probability about?

Three different uses of “ $p$ ”

The null model and extreme results

What makes a result “as extreme or more extreme”?

Check that the reference model is appropriate

Choose the correct tail area

Main Menu

Quick Links

The concept in 60 seconds

What is the probability about?

Three different uses of “pp”

The null model and extreme results

What makes a result “as extreme or more extreme”?

Check that the reference model is appropriate

Choose the correct tail area

Main Menu

Quick Links​

Three different uses of “ $p$ ”

Quick Links