AP®︎ Statistics: Topic 4.10 Flashcards

Inference procedure for comparing μ1 − μ2 using two independent samples or independently assigned treatments when population standard deviations are unknown

Parameter of interest: μ1 − μ2, where μ1 and μ2 are the two population means

Statistic for this procedure: x̄1 − x̄2

Usually H₀: μ1 − μ2 = 0, equivalently H₀: μ1 = μ2

Ha can be μ1 − μ2 ≠ 0, μ1 − μ2 > 0, or μ1 − μ2 < 0, depending on the investigative question

Keep the same order in parameter, sample statistic, and test statistic; reversing order changes the sign and can change the tail for a one-sided test

The two samples or treatment groups must be independent; matched pairs are not analyzed with a two-sample t-test

Data must come from two independent random samples or from a randomized experiment with random assignment to treatments

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

The distribution of x̄1 − x̄2 can be modeled reasonably well by a normal distribution if both population distributions are approximately normal, or both n1 and n2 are at least 30, or if either sample is small, both sample distributions show no strong skewness or outliers

SE = √(s1²/n1 + s2²/n2), the estimated standard deviation of the sampling distribution of x̄1 − x̄2

The AP Statistics two-sample t procedure does not assume equal population variances, so it uses separate sample variances rather than a pooled estimate

t = [(x̄1 − x̄2) − Δ0] / √(s1²/n1 + s2²/n2); for H₀: μ1 − μ2 = 0, t = (x̄1 − x̄2) / √(s1²/n1 + s2²/n2)

Use technology for df; it may be noninteger and falls between min(n1 − 1, n2 − 1) and n1 + n2 − 2

If technology is unavailable, use df = min(n1 − 1, n2 − 1); this is conservative because it generally gives a larger p-value

Probability, assuming H₀ is true, of getting a test statistic as extreme or more extreme than the observed one in the direction of Hₐ; here that means assuming the two population means are equal in context

For Ha: μ1 − μ2 > 0, use area right of t; for Ha: μ1 − μ2 < 0, use area left of t; for Ha: μ1 − μ2 ≠ 0, use both tails beyond ±|t|.

Compare the p-value to α: if p-value ≤ α, reject H₀; if p-value > α, fail to reject H₀.

State in context whether there is or is not convincing statistical evidence about the population means, consistent with Hₐ, referring to the parameters and populations and using non-definitive language

Independent random samples allow generalization to the populations; random assignment in an experiment allows cause-and-effect conclusions about the treatments