AP Stats TPS4e Ch 10.1 Comparing Two Proportions

Sampling distribution of p̂₁ - p̂₂

Choose a SRS of size n₁ from population 1 with proportion of successes p₁ and an independent SRS of size n₂ from the population of successes p₂. Shape: when the four possible np and n(1-p) are all ≥ 10, the sampling distribution is approximately Normal. Center: the mean of the sampling distribution is p̂₁ - p̂₂ = p₁ - p₂. Spread: the standard deviation of the sampling distribution uses √[p̂₁(1-p̂₁)/n₁+p̂₂(1-p̂₂)/n₂], as long as each sample is ≤ 10% of its population

Conditions for testing a hypothesis about p₁ - p₂

Same conditions for a confidence interval for p₁ - p₂ Random: 2 independent SRSs or 2 groups in a randomized experiment 10% rule: n≤N/10 for each sample (only if sampling without replacement) Large n: n₁p̂₁, n₁(1-p̂₁), n₂p̂₂, n₂(1-p̂₂) are all ≥10

Conditions for a confidence interval for p₁ - p₂

Same conditions for testing a hypothesis about p₁ - p₂ Random: 2 independent SRSs or 2 groups in a randomized experiment 10% rule: n≤N/10 for each sample (only if sampling without replacement) Large n: n₁p̂₁, n₁(1-p̂₁), n₂p̂₂, n₂(1-p̂₂) are all ≥10

Two-sample z test for the difference between two proportions p₁-p₂

Suppose the Random, Normal, and Independent conditions are met. To test the hypothesis H₀ : p₁ − p₂ = 0 , first find the pooled proportion of successes in both samples combined. Then compute the z statistic using the given formula. Find the P-value by calculating the probability of getting a z statistic this large or larger in the direction specified by the alternative hypothesis Ha .

Standard Error (SE) of p₁-p₂ for two-sample z interval for a difference between two proportions

The estimated standard deviation of the statistic is given by this formula. Pooled sample proportion is not used here.

Standard Error (SE) of p₁-p₂ for the two-sample z test for the difference between two proportions.

The estimated standard deviation of the statistic is given by this formula. Pooled sample proportion is used here.

Pooled (combined) sample proportion

The overall proportion of successes in the two samples is the count of successes in both samples combined / count of individuals in both samples combined

Two-sample z interval for a difference between two proportions p₁-p₂

When the Random, Normal, and Independent conditions are met, an approximate level C confidence interval for p₁-p₂ is (p₁-p₂) ± z*√[p̂₁(1-p̂₁)/n₁+p̂₂(1-p̂₂)/n₂] where z* is the critical value for the standard Normal curve with area C between -z* and z*.