AP®︎ Statistics: Topic 3.13 Flashcards

Master key terms and definitions for Topic 3.13 of AP Statistics – Carrying Out 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 a Difference Between Population Proportions

Definition

Significance test for whether an observed difference p̂1 − p̂2 provides convincing evidence of a difference between two population proportions

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Two-Sample Z-Test for a Difference Between Population Proportions
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Significance test for whether an observed difference p̂1 − p̂2 provides convincing evidence of a difference between two population proportions

3.13.A3.13.A.1
Pooled Proportion (p̂c, Combined Proportion)
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Estimate of the common population proportion under H₀: p̂c = (x1 + x2)/(n1 + n2) = (n1p̂1 + n2p̂2)/(n1 + n2)

3.13.A3.13.A.1
Null Standard Error for Two Proportions (SE₀)
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Standard error used in the test under H₀: SE₀ = √[p̂c(1 − p̂c)(1/n1 + 1/n2)]

3.13.A3.13.A.1
Two-Proportion Z Test Statistic
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z = (p̂1 − p̂2 − 0)/√[p̂c(1 − p̂c)(1/n1 + 1/n2)]; measures how many null standard errors the observed difference is from 0

3.13.A3.13.A.1
Null Hypothesis for a Two-Proportion Test
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H₀: p1 = p2, equivalently H₀: p1 − p2 = 0

3.13.C3.13.C.1
Alternative Hypotheses for a Two-Proportion Test
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Ha can be p1 > p2, p1 < p2, or p1 ≠ p2; choose from the investigative question before seeing the data

3.13.C3.13.C.3
Large Expected Counts Condition for a Two-Proportion Z-Test
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Using p̂c, all four expected counts must be at least 10: n1p̂c, n1(1 − p̂c), n2p̂c, and n2(1 − p̂c)

3.13.A
P-Value for a Two-Proportion Z-Test
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Probability, assuming H₀ is true, of getting a test statistic at least as extreme as the observed z-statistic in the direction of Hₐ

3.13.B3.13.B.1
Right-Tailed P-Value for a Two-Proportion Z-Test
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For Ha: p1 > p2, p-value = P(Z ≥ zobs)

3.13.A3.13.A.2
Left-Tailed P-Value for a Two-Proportion Z-Test
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For Ha: p1 < p2, p-value = P(Z ≤ zobs)

3.13.A3.13.A.2
Two-Sided P-Value for a Two-Proportion Z-Test
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For Ha: p1 ≠ p2, p-value = P(Z ≤ −|zobs|) + P(Z ≥ |zobs|), or twice the one-tail area beyond |zobs|

3.13.A3.13.A.2
Decision Rule Using α
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If p-value ≤ α, reject H₀; if p-value > α, fail to reject H₀

3.13.C3.13.C.1
Conclusion in Context for a Two-Proportion Test
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Compare p-value with α, state reject H₀ or fail to reject H₀, and conclude in context in terms of Hₐ about the population proportions using non-definitive language

3.13.C3.13.C.3
Order of Groups
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Keep group 1 and group 2 in the same order throughout hypotheses, p̂1 − p̂2, z, p-value, and conclusion; reversing order changes the sign

3.13.A3.13.A.1
Why Pool in a Two-Proportion Test
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Under H₀, p1 = p2, so both samples estimate one common proportion and are pooled to estimate it

3.13.A3.13.A.1
Randomization Condition for a Two-Proportion Z-Test
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Data come from two independent random samples or from a randomized experiment; the two groups must be independent

3.13.A
10% Condition for a Two-Proportion Z-Test
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When sampling without replacement, each sample is no more than 10% of its population: n1 ≤ 0.10N1 and n2 ≤ 0.10N2; not needed for a randomized experiment

3.13.A