AHSS Ch. 6.2 Difference of two proportions
True or False: The SE formula for a confidence interval for a difference of proportions is the same as the SE formula for a hypothesis test for a difference of proportions.
False. The SE formula for a confidence interval for a difference of proportions uses each sample proportion separately. The SE formula for a hypothesis test for a difference of proportions uses the pooled proportion.
ICCCC for a hypothesis test for the difference of two proportions
Identify: H₀: p₁=p₂ Hₐ: p₁≠p₂; p₁>p₂; p₁<p₂ significance level ⍺ Choose: 2-proportion Z-test Check: 1. Data come from 2 independent random samples or from 2 randomly assigned treatments. 2. Check success-failure condition using the pooled proportion. Calculate: Z=(point estimate - null value)/SE of estimate point estimate: the difference of sample proportions p̂₁-p̂₂ SE of estimate: (see image) where p̂ is the pooled proportion null value: assuming H₀ is true, p₁-p₂=0 p-value = (based on the Z-statistic and the direction of Hₐ) Conclude: The p-value is < ⍺, so we reject H₀; there is sufficient evidence that [Hₐ in context]. OR The p-value is > ⍺, so we do not reject H₀; there is not sufficient evidence that [Hₐ in context].
ICCCC for constructing a confidence interval for the difference of two proportions
Identify: the parameter and C% confidence level Choose: 2 proportion Z-interval Check: 1. Data come from 2 independent random samples or from 2 randomly assigned treatments. 2. n₁p̂₁≥10, n₁(1−p̂₁)≥10, n₂p̂₂≥10, and n₂(1−p̂₂)≥10 Calculate: point estimate ± z⋆ × SE of estimate point estimate: the difference of sample proportions p̂₁-p̂₂ Conclude: We are C% confident that the true difference in the proportion of [...] is between ___ and ___. If applicable, draw a conclusion based on whether the interval is entirely above, is entirely below, or contains the value 0.
Conditions for the sampling distribution of p̂₁-p̂₂ to be normal
The difference p̂₁-p̂₂ tends to follow a normal model when 1) each sample proportion separately follows a normal model and 2) the observations and the two samples are independent of each other.
When H₀: p₁-p₂= 0, which proportion do we use in calculating the SE?
When the null hypothesis states that the proportions are equal, we use the pooled sample proportion to estimate the standard error:
Standard deviation for the difference of sample proportions
where p₁ and p₂ represent the population proportions, and n₁ and n₂ represent the sample sizes
