AP®︎ Statistics: Topic 4.5 Flashcards

t = (x̄ − μ₀)/(s/√n)

t = (x̄_d − μ_d,0)/(s_d/√n), usually with μ_d,0 = 0; n is the number of pairs

Df = n − 1

SEx̄ = s/√n

Null distribution for the one-sample t statistic when σ is unknown; if H₀ is true and conditions are met, t has a t-distribution with df = n − 1

Positive t means x̄ > μ₀, negative t means x̄ < μ₀; for paired data, the sign depends on how differences were defined

Probability, assuming H₀ is true, of getting a test statistic as extreme or more extreme than the observed one in the direction of H_a

For H_a: μ > μ₀, p-value = P(T ≥ t_obs)

For H_a: μ < μ₀, p-value = P(T ≤ t_obs)

For H_a: μ ≠ μ₀, p-value = P(|T| ≥ |t_obs|) = 2P(T ≥ |t_obs|)

Use the tail or tails specified by H_a, not the side where the sample mean happened to fall

Use the row for df = n − 1; locate |t_obs| to get the upper-tail area, double it for a two-sided test, and if df is missing use the next smaller df for a conservative p-value

Enter μ₀, x̄, s, n, and the alternative, or use raw data; report the test statistic t, the p-value, and df

α, the cutoff for deciding whether the evidence against H₀ is strong enough; it should be chosen before examining the data

If p-value ≤ α, reject H₀; if p-value > α, fail to reject H₀.

A result is statistically significant at level α if p-value ≤ α.

Conclude the data provide convincing statistical evidence for the claim in H_a, not proof that H_a is true

Conclude the data do not provide convincing evidence for H_a, not that H₀ is true

Design with dependent observations within each pair; calculate a difference for each pair and analyze those differences with a one-sample t-test

μ_d, the population mean of the paired differences

Define d consistently, such as first measurement − second measurement; reversing the order changes the signs of x̄_d, t, and any directional H_a

Significance test for a population mean μ when σ is unknown, or for a matched-pairs population mean difference μ_d using the pair differences; uses t = (x̄ − μ₀)/(s/√n) with df = n − 1

State the decision and a non-definitive conclusion consistent with H_a, naming the population, response variable, and population mean or mean paired difference; use wording like “there is convincing evidence” or “there is not convincing evidence,” not “proved” or “definitely is.”

Data come from a random sample or a randomized experiment

When sampling without replacement, n/N ≤ 0.10 so observations can be treated as approximately independent

If n ≥ 30, the t procedure is generally okay unless the population is extremely skewed; if n < 30, the sample data should be free from strong skewness and outliers