AP Statistics Topic 3.12: Setting Up a Test for the Difference Between Two Population Proportions

01 · Start with a population question

The concept in 60 seconds

School A wants to know whether its students are more likely to prefer digital study notes than students at School B. Independent random samples show $124$ of $200$ students at A and $69$ of $150$ students at B prefer digital notes.

The sample proportions are $0.62$ and $0.46$ , a difference of $0.16$ , or $16$ percentage points. Could a difference like this arise from sampling variation when the population proportions are actually equal? That is the question a test can address.

The setup has four jobs: define the population parameters, choose a two-sample $z$ -test for a difference in proportions, state $H_{0}$ and $H_{a}$ , and justify the conditions. The test statistic, $p$ -value and final decision follow in Topic 3.13.

Visual guide 1: connect the question, parameters and plan

Question set in advanceIs A’s rate higher?

Same binary preference outcome, $2$ school populations.

Unknown target $p_{1}-p_{2}$

$p_{1}$ is A’s population preference proportion; $p_{2}$ is B’s. Keep A minus B throughout.

Test setup $H_0:p_1-p_2=0$

$H_a:p_1-p_2\gt0$ . Justify independent random samples, each $10\%$ check and $4$ pooled expected counts.

The observed $0.16$ difference is sample evidence. Hypotheses concern the unknown population difference, and the plan must be justified before interpreting a test result.

Here the research question was “Is A higher?” before the results were examined. Therefore, with A minus B as our order:

$H_0:p_1-p_2=0$ — the population preference proportions are equal.

$H_a:p_1-p_2\gt0$ — School A’s population preference proportion is higher.

Quick check: does the sample difference of

0.16

prove that A’s population proportion is higher?

No. It is observed sample evidence. A justified test evaluates how unusual that evidence would be under $H_{0}$ ; the setup alone does not establish a population difference.

02 · Name who and what each proportion describes

Define the $2$ population parameters

A strong parameter definition names the population, the response category counted as a success and the group order. “Success” is just the category we count; it need not be a desirable outcome.

$p_{1}$ : School A

The proportion of all students at School A who prefer digital study notes.

It is an unknown population value. Its sample estimate is $\hat{p}_{1}=\frac{124}{200}=0.62$ .

$p_{2}$ : School B

The proportion of all students at School B who prefer digital study notes.

Its sample estimate is $\hat{p}_{2}=\frac{69}{150}=0.46$ . Use the same preference question and success definition in both groups.

Our target is $p_{1}-p_{2}$ , the population difference A minus B. The sample statistic $\hat{p}_{1}-\hat{p}_{2}$ estimates this difference. Hypotheses concern the population parameters $p_{1}$ and $p_{2}$ , not the observed sample proportions.

The school study: keep population parameters, observed statistics and null-model quantities separate.
Symbol	Meaning	School value or role
$p_{1}$ , $p_{2}$	Population proportions with the same digital-notes preference	Unknown; A and B, respectively
$p_{1}-p_{2}$	Population difference being tested	A minus B
$x_{1}$ , $x_{2}$	Observed success counts	$124$ and $69$
$n_{1}$ , $n_{2}$	Sample sizes	$200$ and $150$
$N_{1}$ , $N_{2}$	Population sizes	$10{,}000$ and $8{,}000$
$\hat{p}_{1}$ , $\hat{p}_{2}$	Observed sample proportions	$0.62$ and $0.46$
$\hat{p}_{1}-\hat{p}_{2}$	Observed sample difference	$0.16=16\,\text{percentage points}$
$\hat{p}_{c}$	Combined sample proportion used under $H_{0}$	$\frac{193}{350}\approx 0.5514$
$H_{0}$	Equality of the population rates	$p_{1}-p_{2}=0$
$H_{a}$	Departure chosen from the research question	$p_{1}-p_{2} > 0$

For a randomized experiment

Define the parameters as the underlying success probabilities or success proportions under the $2$ treatments in the population or experimental setting of interest. Name each treatment and the same response outcome. Do not automatically claim that volunteer participants represent every student or every customer.

Quick check: what is missing from “

p_{1}

is the proportion who said yes”?

The definition does not identify the population or explain what “yes” means. Write, for example, “ $p_{1}$ is the proportion of all School A students who prefer digital study notes.”

03 · Let the research question choose the direction

Write the null and alternative hypotheses

The usual AP two-proportion $z$ -test here investigates no population difference as the null. You can write this in either equivalent form:

$H_0:p_1=p_2$ or $H_0:p_1-p_2=0$ .

The alternative describes the departure of interest. First define your order, then translate the original research question:

Choose the alternative for group 1 minus group 2. The null is $p_{1}-p_{2}=0$ in every row.
Original question	Alternative hypothesis	Direction
Is group 1’s success proportion higher?	$H_{a}: p_{1}-p_{2} > 0$	One-sided: higher
Is group 1’s success proportion lower?	$H_{a}: p_{1}-p_{2} < 0$	One-sided: lower
Are the population proportions different?	$H_{a}: p_{1}-p_{2} \ne 0$	Two-sided: either direction

Visual guide 2: the alternative chooses the evidence direction

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Setting Up a Test for the Difference Between Two Population Proportions

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

The concept in 60 seconds