6.10 Setting Up a Test for the Difference of Two Population Proportions

Setting up a two-sample z-test for a difference of two proportions means writing your hypotheses, defining what each proportion represents, and checking that your conditions are met before you calculate anything. The null hypothesis says the two population proportions are equal, and the big change here is that you check normality using a pooled (combined) proportion instead of separate ones.

Why This Matters for the AP Statistics Exam

This topic is part of Unit 6, which carries a good share of the exam. Comparing two proportions shows up when a question asks whether two groups truly differ on some categorical outcome, like the proportion of two populations that succeed, agree, or respond a certain way.

On the exam, you may need to:

• Recognize that a comparison of two groups on a categorical variable calls for a two-sample z-test for a difference of proportions.
• Write correct hypotheses and define parameters clearly.
• Verify independence and normality conditions, including the pooled proportion check.

Getting the setup right matters because the rest of the test (calculating the statistic, finding the p-value, and concluding) depends on it. Carrying out the full test comes next in 6.11.

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Key Takeaways

• The null hypothesis is always H₀: p₁ = p₂ (equivalently p₁ - p₂ = 0). The alternative is one-sided (p₁ > p₂ or p₁ < p₂) or two-sided (p₁ ≠ p₂).
• Define p₁ and p₂ in context so the reader knows exactly which populations you are comparing.
• The correct procedure is a two-sample z-test for a difference of two population proportions.
• Check independence: two independent random samples (or random assignment in an experiment), plus the 10% condition when sampling without replacement.
• Check normality with a pooled proportion p̂c, then confirm n₁p̂c, n₁(1-p̂c), n₂p̂c, and n₂(1-p̂c) are all large enough (commonly at least 10).
• Use meaningful subscripts so it is clear which proportion goes with which group.

Hypotheses and Parameters

The first step is writing your hypotheses. The null hypothesis always sets the two population proportions equal, and the alternative says one is greater than, less than, or not equal to the other.

• Null: H₀: p₁ = p₂, or equivalently H₀: p₁ - p₂ = 0
• One-sided alternative: Hₐ: p₁ > p₂ or Hₐ: p₁ < p₂
• Two-sided alternative: Hₐ: p₁ ≠ p₂

Just as important, define what p₁ and p₂ represent. State the populations in words so the reader knows exactly what you are comparing. The null hypothesis sets the difference to 0, which is the "no difference or effect" situation.

Conditions

You also have to check the conditions for inference. The three checks are similar to the ones for the confidence interval, with one change in the normal check.

(1) Random

Both samples need to come from random samples (or random assignment in an experiment). Without randomization, your results suffer from bias and you cannot generalize to the populations.

(2) Independence

Check that the data come from two independent random samples or a randomized experiment. When sampling without replacement, also confirm the 10% condition: n₁ ≤ 10% N₁ and n₂ ≤ 10% N₂. In a randomized experiment, random assignment of treatments is what makes the groups independent.

(3) Normal

For proportions, you check normality using the Large Counts idea: expected successes and failures should be large enough (commonly at least 10). The change for a two-sample test is that you assume H₀ is true (p₁ = p₂), so you combine both samples into one pooled proportion:

p̂c = (n₁p̂₁ + n₂p̂₂)/(n₁ + n₂)

Then check that all four of these are large enough (commonly at least 10):

• n₁p̂c
• n₁(1-p̂c)
• n₂p̂c
• n₂(1-p̂c)

You pool because the null hypothesis assumes the two proportions are equal. Under that assumption, combining the samples gives a single best estimate of the common proportion, which you use to check the shape condition.

Example

Suppose MJ made 836 of 1623 shots and Lebron made 622 of 1493 shots, and you want to test whether their shooting proportions differ. Here is how the setup looks for a two-sample z-test.

Hypotheses and Parameters

• H₀: p_MJ = p_L (their population shot-making proportions are equal)
• Hₐ: p_MJ ≠ p_L (their population shot-making proportions are different)

Let p_MJ be the true proportion of shots MJ makes and p_L be the true proportion of shots Lebron makes. Using meaningful subscripts like MJ and L keeps it clear which proportion matches which group.

Conditions

• Random: The problem does not state the shots were randomly sampled, so this is an assumption you would have to make and note as a limitation.
• Independent: It is reasonable that each player took far more than 10 times the number of shots in the sample, so the 10% condition is satisfied and the two samples are independent of each other.
• Normal: First find the pooled proportion.

p̂c = (836 + 622)/(1623 + 1493) ≈ 0.468

Now check Large Counts using p̂c ≈ 0.468 (so 1 - p̂c ≈ 0.532):

• 1623(0.468) ≈ 760 ≥ 10 ✔️
• 1623(0.532) ≈ 863 ≥ 10 ✔️
• 1493(0.468) ≈ 699 ≥ 10 ✔️
• 1493(0.532) ≈ 794 ≥ 10 ✔️

All conditions check out, so the setup is complete and you are ready to calculate the test statistic and find the p-value (that is the next topic, 6.11).

How to Use This on the AP Statistics Exam

Free Response

• Name the procedure: a two-sample z-test for a difference of two population proportions.
• Write H₀: p₁ = p₂ and the correct alternative based on the question.
• Define p₁ and p₂ in context, with clear subscripts.
• Show all three conditions: random, independent (including the 10% check), and normal with the pooled proportion.
• Clear notation and labeled work are important for communicating your setup, even before you calculate anything.

MCQ

• Be ready to pick the correct hypotheses or the correct procedure from a list.
• Watch for the pooled proportion in the normal check; that is the detail that separates one-sample and two-sample proportion tests.

Common Trap

• Forgetting to pool. For the difference of two proportions, the normality check uses the combined p̂c, not separate p̂₁ and p̂₂.

Common Misconceptions

• The null hypothesis is not about a specific value like 0.5 for each group. It only says the two proportions are equal, which is the same as saying their difference is 0.
• "Failing to reject" does not prove the proportions are equal. A test can only give evidence for a difference, not prove there is none.
• The 10% condition is about independence within each sample (sampling without replacement), not about the normal condition. Keep those two checks separate.
• The pooled proportion is only for the test setup under H₀. The confidence interval for a difference of proportions does not pool; it uses the separate sample proportions.
• A small p-value will come later, but it shows the difference would be unusual if H₀ were true. It does not, by itself, measure how big the difference is.

Related AP Statistics Guides

• Unit 6 Overview: Inference for Categorical Data: Proportions
• 6.2 Constructing a Confidence Interval for a Population Proportion
• 6.1 Introducing Statistics: Why Be Normal?
• 6.4 Setting Up a Test for a Population Proportion
• 6.3 Justifying a Claim Based on a Confidence Interval for a Population Proportion
• 6.5 Interpreting p-Values