Comparing Two Population Proportions: Inference and Hypothesis Testing - Prof. Brett D. Hu, Study notes of Data Analysis & Statistical Methods

Download Comparing Two Population Proportions: Inference and Hypothesis Testing - Prof. Brett D. Hu and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity! 21 October Comparing 2 population proportions We want to make inferences about π1 – π2, where π1 and π2 come from different populations Take independent samples Find p1, the point estimate for π1 Find p2, the point estimate for π2 So p1 – p2 is the point estimate for π1- π2 Sampling Distribution When the samples are independently taken and n is large, π1, π2 not extreme, following hold n1π1 ≥ 10 n2π2 ≥ 10 n1(1 – π1) ≥ 10 n2(1 – π2) ≥ 10 p1 – p2 ~ N (π1 – π2, π1(1−π1) n1 + π 2(1−π2) n2 Hypotheses Upper Tail: Is π1 larger than π2? H0: π1- π2 ≤ 0 HA: π1 – π2> 0 OR H0: π1 ≤ π2 HA: π1 > π2 Lower Tail: Is π1 smaller than π2? H0: π1 – π2 ≥ 0 HA: π1- π2 < 0 OR H0: π1 ≥ π2 HA: π1 < π2 Two Tail: Is π1different from π2? H0: π1- π2 = 0 HA: π1- π2≠ 0 OR H0: π1= π2 HA: π1≠ π2 When we perform a hypothesis test, we assume H0 is true. So here, we are assuming π1= π2 When π1 = π2, we can combine the two populations into one where the true proportion is π = π1 = π2 So we can combine our samples into one sample of size n1 + n2, from which we get an estimate of the overall population proportion (π), denoted pc or p. Called combined or pooled population That is, pc is the sample proportion from the combined sample of size n1 + n2 pc = p = n1 p1+n2 p2 n1+n2 = ¿ successes∈sample1+¿ successes∈sample 2 n1+n2 Since we assume π1 = π2 = π, we can assume p1 ≈ p2≈ pc (Theoretically, p1 would = p2, but won’t be in every case. We use pc as an estimate for both) If sampling distribution assumptions hold, then