AP®︎ Statistics: Topic 4.6 Flashcards

The distribution of x̄1 − x̄2 from all possible independent random samples of sizes n1 and n2

μx̄1−x̄2 = μ1 − μ2

σx̄1−x̄2 = √(σ1²/n1 + σ2²/n2) for independent samples

Keep sample and population differences in the same order; reversing the order changes only the sign, not the standard deviation

For samples, use two independent random samples; for an experiment, randomly assign treatments to independent groups

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

For randomized experiments, use random assignment to independent treatment groups; the 10% condition is not required

x̄1 − x̄2 ~ N(μ1 − μ2, √(σ1²/n1 + σ2²/n2)); if justified by large samples rather than normal populations, this model is approximate

z = [(x̄1 − x̄2) − (μ1 − μ2)] / √(σ1²/n1 + σ2²/n2), or for a cutoff c: z = [c − (μ1 − μ2)] / √(σ1²/n1 + σ2²/n2)

Interpret as a probability about repeated samples and the resulting values of x̄1 − x̄2, not about a fixed population mean difference

This model applies only to independent groups; matched pairs are analyzed as one sample of within-pair differences

The sampling distribution of x̄1 − x̄2 is normal if both population distributions are normal; if not, it is approximately normal when n1 ≥ 30 and n2 ≥ 30