AP®︎ Statistics: Topic 3.9 Flashcards

Statistic p̂1 − p̂2, used to estimate the difference in population proportions p1 − p2

Parameter giving the true difference in success proportions for population 1 minus population 2

μ_p̂1−p̂2 = p1 − p2

σ_p̂1−p̂2 = sqrt[p1(1 − p1)/n1 + p2(1 − p2)/n2].

For samples: use two independent random samples. For experiments: randomly assign treatments to the experimental units

When sampling without replacement, each sample must be no more than 10% of its own population: n1 ≤ 0.10N1 and n2 ≤ 0.10N2

Approximate normality requires all four expected counts to be at least 10: n1p1 ≥ 10, n1(1 − p1) ≥ 10, n2p2 ≥ 10, and n2(1 − p2) ≥ 10

Distribution of the values of p̂1 − p̂2 over repeated independent random samples of the same sizes, or repeated random assignment in an experiment

If conditions are met, p̂1 − p̂2 is approximately normal with mean p1 − p2 and standard deviation sqrt[p1(1 − p1)/n1 + p2(1 − p2)/n2].

z = [d − (p1 − p2)] / sqrt[p1(1 − p1)/n1 + p2(1 − p2)/n2].

Keep the same group first in both p1 − p2 and p̂1 − p̂2; reversing order changes the sign of the difference but not the standard deviation

A probability about p̂1 − p̂2 is the long-run proportion of repeated samples or random assignments that produce that result, not a probability about fixed p1 and p2

This procedure does not apply to paired or matched observations because the two sample proportions are associated, not independent