AP®︎ Statistics: Topic 3.7 Flashcards

Hypothesis test for one population proportion p using z = (p̂ - p₀)/√(p₀(1 - p₀)/n) when conditions are met

H₀: p = p₀; if a null is first written with an inequality, the test is carried out at the boundary value p = p₀.

Hₐ: p > p₀, Hₐ: p < p₀, or Hₐ: p ≠ p₀; determines which tail or tails count as evidence

The true proportion in the population with the response of interest; define the population and what counts as a success in context

Randomization; 10% condition if sampling without replacement; normality under H₀: np₀ ≥ 10 and n(1 - p₀) ≥ 10

When sampling without replacement from a finite population, require N ≥ 10n so observations are approximately independent

Under H₀, expected successes and failures must both be at least 10: np₀ ≥ 10 and n(1 - p₀) ≥ 10

p̂ = x/n, where x is the number of successes in the sample

√(p₀(1 - p₀)/n), the standard deviation of p̂ assuming H₀ is true

z = (p̂ - p₀)/√(p₀(1 - p₀)/n); measures how many null standard deviations p̂ is from p₀.

Distribution of the test statistic assuming H₀ is true; approximately standard normal when conditions are met

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

For Hₐ: p > p₀, p-value = P(Z ≥ z_obs)

For Hₐ: p < p₀, p-value = P(Z ≤ z_obs)

For Hₐ: p ≠ p₀, p-value = P(Z ≤ -|z_obs|) + P(Z ≥ |z_obs|)) = 2P(Z ≥ |z_obs|)

α, the predetermined probability of rejecting H₀ when H₀ is true

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

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

Conclude there is convincing statistical evidence to support Hₐ.

Conclude there is not convincing statistical evidence to support Hₐ; this does not prove H₀ true

State the conclusion in context about the population proportion, consistent with Hₐ, using non-definitive language such as 'there is convincing statistical evidence' or 'there is not convincing statistical evidence.'

State: define p and hypotheses. Plan: identify a one-sample z-test and check conditions. Do: calculate p̂, z, and p-value. Conclude: compare p-value to α, make the decision, and state the conclusion in context

A small p-value shows evidence against H₀, not whether the difference is important in real-world terms

Data should come from a random sample of the population