DATASCI 6.2

What parameter are we trying to estimate in two-proportion problems?

The difference in population proportions: p1−p2

What is the point estimate for p1−p2?

p^1−p^2

What distribution is used for inference with two proportions?

A normal (z) distribution, if conditions are met

What are the two main conditions for using a normal model?

Independence (within and between groups) and Success-failure condition (for both groups)

What does "independence (extended)" mean?

Observations are independent within each sample AND the two samples are independent of each other

When is independence usually satisfied?

When data come from: Two random samples OR a randomized experiment

What is the success-failure condition for two proportions?

Each group must have: ≥10 successes and ≥10 failures

What does standard error measure in this context?

The variability of p^1−p^2

What is the general idea of the SE formula for two proportions?

It combines variability from both groups

What is the point estimate used in a 2-proportion CI?

p^1−p^2

What are the 4 steps to constructing a CI?

Prepare → Check → Calculate → Conclude

How do you interpret a CI for p1−p2?

As a range of plausible values for the true difference in population proportions

What is the null hypothesis in most 2-proportion tests?

H0:p1−p2=0 (no difference) or p1=p2

What does the alternative hypothesis represent?

A difference between proportions (≠,>,

When do we use the pooled proportion?

ONLY when H0:p1−p2=0

What does the pooled proportion represent?

A combined estimate of the common proportion assuming p1=p2

Why do we pool proportions in hypothesis testing?

Because the null hypothesis assumes both groups have the same true proportion

Is pooled proportion used in confidence intervals?

No, only in hypothesis tests when H0=0

What if the null hypothesis is p1−p2=0.1?

Do NOT use pooled proportion

In non-zero null cases, what proportions are used?

p^1p^1 and p^2p^2 separately

What's the biggest difference between CI and hypothesis testing here?

CI → no pooling, Hypothesis test → pooling (if H0=0)

What must always be checked before inference?

Conditions (independence + success-failure)

Why do we check success-failure separately for each group?

Because each sample has its own variability

Summarize the full process for 2-proportion inference

Identify p^1,p^2, Check conditions (both groups), Choose method: CI → no pooling, HT → pool if H0=0, Compute SE and z, Interpret in context