When you compare two groups, the difference in sample proportions p̂₁ − p̂₂ changes from sample to sample. Set two true proportions and two sample sizes, take many pairs of random samples, and compare the simulated differences with the normal model.
Set the two true population proportions and the two sample sizes. Each simulated trial takes an independent random sample from each population and records the difference in sample proportions.
The sampling distribution of p̂₁ − p̂₂ has mean μ = p₁ − p₂. That makes p̂₁ − p̂₂ an unbiased estimator of the difference in population proportions.
For independent samples, σ² = p₁(1 − p₁)/n₁ + p₂(1 − p₂)/n₂. You add the two variances, then take the square root. The variances add even though you subtract the proportions.
The shape is approximately normal when n₁p₁, n₁(1 − p₁), n₂p₂, and n₂(1 − p₂) are all at least 10.
The samples must be independent of each other (two separate random samples or random assignment), and each sample should be at most 10% of its population when sampling without replacement.
σ shrinks as n₁ and n₂ grow. To cut σ in half you need about four times as many observations in each group.
With the normal model, find P(p̂₁ − p̂₂ > c) by computing z = (c − μ)/σ and using a normal table or normalcdf.
The sampling distribution of p̂₁ − p̂₂, and what each condition checks.
| Feature | Formula or rule | Condition needed |
|---|---|---|
| Mean | μ = p₁ − p₂ | None: always true for random samples |
| Standard deviation | σ = √( p₁(1 − p₁)/n₁ + p₂(1 − p₂)/n₂ ) | Independent samples; each n ≤ 10% of its population |
| Shape | Approximately normal | n₁p₁, n₁(1 − p₁), n₂p₂, n₂(1 − p₂) all ≥ 10 |
| z-score | z = ((p̂₁ − p̂₂) − (p₁ − p₂)) ÷ σ | Normal shape |
| Bigger samples | σ gets smaller; center does not move | — |
| Equal populations | μ = 0, but individual differences still vary | — |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 3.9. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.