AP®︎ Statistics: Topic 3.12 Flashcards

Master key terms and definitions for Topic 3.12 of AP Statistics – Setting Up a Test for the Difference Between Two Population Proportions to help you prep for quizzes and the AP exam.

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Term

Two-Sample Z-Test for the Difference Between Two Population Proportions

Definition

Use to test whether two population proportions differ for a binary categorical outcome in two independent populations or independently formed treatment groups

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Two-Sample Z-Test for the Difference Between Two Population Proportions
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Use to test whether two population proportions differ for a binary categorical outcome in two independent populations or independently formed treatment groups

3.12.A3.12.A.1
Parameter for a Difference in Two Population Proportions
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p₁ - p₂, where p₁ and p₂ are the true population proportions of success for the two populations or treatments

3.12.A3.12.A.2
Complete Parameter Definition
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Define each parameter by naming the population or treatment, the success category, and that it is a population proportion, not a sample proportion

3.12.A3.12.A.2
Order of Parameters
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The order in p₁ - p₂ determines the sign of the difference and the direction of any one-sided alternative

3.12.B3.12.B.1
Hypotheses Use Population Proportions, Not Sample Proportions
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Write hypotheses about p₁ and p₂, not p̂₁ and p̂₂, because the test is about unknown population parameters

3.12.B3.12.B.1
Null Hypothesis for Two Proportions
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H₀: p₁ = p₂, equivalently H₀: p₁ - p₂ = 0; it states no difference between population proportions

3.12.B3.12.B.1
Alternative Hypotheses for Two Proportions
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Hₐ: p₁ > p₂, Hₐ: p₁ < p₂, or Hₐ: p₁ ≠ p₂; equivalently, write them as p₁ - p₂ > 0, p₁ - p₂ < 0, or p₁ - p₂ ≠ 0

3.12.B3.12.B.1
Choose Alternative from the Research Question
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Higher/more likely/increases → >; lower/less/decreases → <; different/not the same/has an effect → ≠; choose the direction before looking at the sample results

3.12.B3.12.B.1
Equality in the Null Hypothesis
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For this test, the null is ordinarily written with equality: p₁ = p₂ or p₁ - p₂ = 0

3.12.B3.12.B.1
Randomization Condition for a Two-Proportion Z-Test
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Data must come from two independent random samples or from a randomized experiment with random assignment to independently formed treatment groups

3.12.C3.12.C.1.i
10% Condition for a Two-Proportion Z-Test
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When sampling without replacement, check separately that n₁ ≤ 0.10N₁ and n₂ ≤ 0.10N₂.

3.12.C3.12.C.1.ii
When the 10% Condition Is Unnecessary
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It is not needed for a randomized experiment or when sampling is effectively with replacement

3.12.C3.12.C.1.ii
Normality (Large-Counts) Condition for a Two-Proportion Z-Test
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Using the pooled proportion p̂_c under H₀, all four expected counts must be at least 10: n₁p̂_c, n₁(1 - p̂_c), n₂p̂_c, and n₂(1 - p̂_c)

3.12.C3.12.C.1.iii
Meaning of the Four Expected Counts
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They are the expected successes and failures in each group under the null hypothesis of equal population proportions

3.12.C3.12.C.1.iii
Independent Groups Requirement
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The two groups must be independent; this procedure is not for paired data, matched pairs, or repeated measurements on the same individuals

3.12.C3.12.C.1.i
10% Condition vs. Large-Counts Condition
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The 10% condition supports approximate independence when sampling without replacement; the large-counts condition supports the normal approximation

3.12.C3.12.C.1
Pooled Proportion
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p̂_c = (x₁ + x₂) / (n₁ + n₂) = (n₁p̂₁ + n₂p̂₂) / (n₁ + n₂); used because H₀ assumes p₁ = p₂, so both groups share one common population proportion

3.12.C3.12.C.1.iii