Setting Up a Two-Proportion Test — AP® Statistics

Part 1 · Hook

Why this matters

A city hands out a new recycling bin to 160 randomly chosen households and leaves 140 with the old one. A month later, 78% of the new bin households sort correctly against 65% of the others. Before anyone runs the numbers, the test has to be set up: what exactly is being claimed, which procedure fits, and are its conditions met?

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Which is a correct null hypothesis for one proportion?

$H_0: p = 0.4$
$H_0: \hat{p} = 0.4$
$H_0: p > 0.4$
$H_0: \hat{p} > 0.4$

Show the answer

A null hypothesis is about the parameter, with an equals sign.

Correct: $H_0: p = 0.4$ :
$H_0: \hat{p} = 0.4$ :
$H_0: p > 0.4$ :
$H_0: \hat{p} > 0.4$ :

2. For a test of $H_0: p = 0.4$ with $n = 50$ , which large counts check is right?

$50(0.4) = 20$ and $50(0.6) = 30$ are at least 10
$n = 50$ is at least 30
$50\hat{p}$ is at least 10
$50(0.5) = 25$ is at least 10

Show the answer

A test checks large counts with the null value $p_0$ .

Correct: $50(0.4) = 20$ and $50(0.6) = 30$ are at least 10:
$n = 50$ is at least 30:
$50\hat{p}$ is at least 10:
$50(0.5) = 25$ is at least 10:

Part 4 · See it

See it first

The two sample proportions and the pooled proportion

p_c = \frac{216}{300} = 0.72

, the estimate of the common proportion if

H_0

is true. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

The question compares two groups on a yes or no outcomethe parameter is $p_1 - p_2$ , and the procedure is a two-sample z test
The null hypothesis says there is no difference $H_0: p_1 - p_2 = 0$ , and $H_a$ says $>$ , $<$ or $\ne$ as the question asks, chosen before seeing the data
If $H_0$ is true, both samples come from populations with one common proportionestimate it by pooling: $p_c$ $= \frac{x_1 + x_2}{n_1 + n_2}$ , total successes over total sample size
The test is computed as if $H_0$ were truecheck large counts with $p_c$ : $n_1 p_c$ , $n_1(1 - p_c)$ , $n_2 p_c$ , $n_2(1 - p_c)$ each at least 10, plus random and 10%

Part 6 · Key ideas

Key ideas

Hypotheses: $H_0: p_1 - p_2 = 0$ and $H_a: p_1 - p_2 > 0$ , $< 0$ or $\ne 0$ , with each $p$ defined as a true proportion in context.
Pooled proportion: $p_c$ $= \frac{x_1 + x_2}{n_1 + n_2}$ , the combined successes over the combined sample size. Never the average of $\hat{p}_1$ and $\hat{p}_2$ .
Conditions: two independent random samples or randomly assigned groups; 10% for each sample; large counts with $p_c$ .
Name it in full: "a two-sample z test for $p_1 - p_2$ ".

Part 7 · Misconception

A common mistake

The wrong idea: The pooled proportion is the average of the two sample proportions.

What actually happens: Pool the counts, not the proportions: add the successes and divide by the total sample size. With unequal sample sizes the two answers differ, and only the pooled counts give the right estimate.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

New recycling bins

A city wants to know whether a new bin design increases the share of households that sort their recycling correctly. It randomly assigns 300 volunteer households: 160 get the new bin and 140 keep the old one. After a month, inspectors record which households sorted correctly. (Invented data.)

Households that sorted correctly
Bin	Sorted correctly	Households	Sample proportion
New	125	160	0.781
Old	91	140	0.650

1. Which hypotheses match the city's question?

$H_0: p_N - p_O = 0$ , $H_a: p_N - p_O > 0$
$H_0: p_N - p_O > 0$ , $H_a: p_N - p_O = 0$
$H_0: p_N - p_O = 0$ , $H_a: p_N - p_O \ne 0$
$H_0: \hat{p}_N - \hat{p}_O$ $= 0$ , $H_a: \hat{p}_N - \hat{p}_O > 0$

Show the answer

The null is no difference; the alternative is the direction of the research question.

Correct: $H_0: p_N - p_O = 0$ , $H_a: p_N - p_O > 0$ : Right: the null says no difference; the alternative says the new bin is better, the direction the city asked about.
$H_0: p_N - p_O > 0$ , $H_a: p_N - p_O = 0$ : The null hypothesis is the "no difference" claim, with an equals sign; the alternative is what we look for evidence of.
$H_0: p_N - p_O = 0$ , $H_a: p_N - p_O \ne 0$ : The city asked whether the new design increases correct sorting, a one-sided question. Choose the direction before seeing the data.
$H_0: \hat{p}_N - \hat{p}_O$ $= 0$ , $H_a: \hat{p}_N - \hat{p}_O > 0$ : Hypotheses are about parameters. We know the sample proportions differ; the question is about the true proportions.

2. How should $p_N$ be defined in context?

The proportion of the 160 new bin households in this study that sorted correctly
The chance that a household chosen at random picks the new bin over the old one
The number of households in the study that sorted correctly with the new bin
The true share of such households that would sort correctly with the new bin

Show the answer

Define each parameter as a true proportion, in context.

The proportion of the 160 new bin households in this study that sorted correctly: That is $\hat{p}_N = 0.781$ , a statistic we already know.
The chance that a household chosen at random picks the new bin over the old one: Households did not choose; the city assigned the bins at random.
The number of households in the study that sorted correctly with the new bin: A proportion is a fraction of a group, not a count.
Correct: The true share of such households that would sort correctly with the new bin: Right: in an experiment, the parameter is the proportion of subjects like these who would succeed with that treatment.

3. What is the pooled proportion $p_c$ for this test? Give three decimal places.

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

$p_c$ $= \frac{125 + 91}{160 + 140}$ $= \frac{216}{300}$ $= 0.72$ .

Answer: 0.72

4. Which statement correctly checks the large counts condition for this test?

$160(0.5) = 80$ and $140(0.5) = 70$ are each at least 10
Both $\hat{p}_N$ and $\hat{p}_O$ are greater than 0.10
$160(0.72) = 115.2$ , $160(0.28) = 44.8$ , $140(0.72) = 100.8$ and $140(0.28) = 39.2$ are each at least 10
The total of 300 households is at least 30

Show the answer

For the test, $n_1 p_c$ , $n_1(1 - p_c)$ , $n_2 p_c$ , $n_2(1 - p_c)$ are each at least 10.

$160(0.5) = 80$ and $140(0.5) = 70$ are each at least 10: Nothing in the problem is 0.5. The null hypothesis says the proportions are equal, and $p_c$ estimates that common value.
Both $\hat{p}_N$ and $\hat{p}_O$ are greater than 0.10: That is not the condition. Each group needs at least 10 expected successes and failures.
Correct: $160(0.72) = 115.2$ , $160(0.28) = 44.8$ , $140(0.72) = 100.8$ and $140(0.28) = 39.2$ are each at least 10: Right: a test assumes $p_N = p_O$ , so the four counts use the pooled proportion $p_c = 0.72$ .
The total of 300 households is at least 30: Comparing $n$ with 30 is a rule for means. Check counts against 10.

5. A researcher looks at her data, sees that $\hat{p}_1 > \hat{p}_2$ , and then writes $H_a: p_1 - p_2 > 0$ . What is wrong?

The direction of $H_a$ must come from the question, not the data
$H_a$ should say $p_1 - p_2 = 0$
$H_a$ should be written about the sample proportions, $\hat{p}_1 - \hat{p}_2$
One-sided alternatives are not allowed for two proportions

Show the answer

Set hypotheses from the question, before the data.

Correct: The direction of $H_a$ must come from the question, not the data: Right: choosing a one-sided test because of the sample result makes evidence look stronger than it is.
$H_a$ should say $p_1 - p_2 = 0$ : The equality belongs in $H_0$ .
$H_a$ should be written about the sample proportions, $\hat{p}_1 - \hat{p}_2$ : Hypotheses are about parameters, never sample statistics.
One-sided alternatives are not allowed for two proportions: One-sided alternatives are fine when the question has a direction.

6. A clinic randomly assigns 30 patients to each of two treatments for a rare reaction. In all, 6 of the 60 patients have the reaction. Why is the large counts condition not met for a two-sample z test?

$30(0.90) = 27$ is more than 10
The groups have the same size, 30
Random assignment was used instead of random sampling
$p_c = 0.10$ , so $30(0.10) = 3$ is less than 10 in each group

Show the answer

Each of the four counts with $p_c$ must reach 10.

$30(0.90) = 27$ is more than 10: That count passes. It is the other count, $30(0.10) = 3$ , that fails.
The groups have the same size, 30: Equal groups are fine. The small number of reactions expected in each group is the problem.
Random assignment was used instead of random sampling: Random assignment satisfies the random condition. The counts are the problem.
Correct: $p_c = 0.10$ , so $30(0.10) = 3$ is less than 10 in each group: Right: with so few reactions overall, each group expects only 3, too few for a normal model.

Part 9 · Summary

Summary

A two-sample z test for

p_1 - p_2

starts from

H_0: p_1 - p_2 = 0

and an alternative chosen from the question. Because the test assumes

H_0

, it pools the samples,

p_c

= \frac{x_1 + x_2}{n_1 + n_2}

, and checks large counts with

p_c

, along with two independent random samples (or random assignment) and the 10% condition.

Part 10 · Up next

What comes next

Part 11 · Connections