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4/5/11 Lecture 18 1 STOR 155 Introductory Statistics Lecture 18: Inference for a single proportion Section 8.1 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL4/5/11 Lecture 18 2 Normal Approximation for Counts and Proportions • Let X ~ B(n, p) and • If n is large, then • Rule of Thumb: np  10, n(1 - p)  10. ./ˆnXp )./)1( ,( approx. is ˆn-pppNp4/5/11 Lecture 18 3 Confidence Interval for p • Expression: where the margin of error – Assumption: n is large – Confidence level C determines m = zãpê(1 àpê)=np[ pê à m ; pê + m ]zã4/5/11 Lecture 18 4 Hypothesis Testing for p • For a hypothesized value , we want to test versus some alternative (1-sided or 2-sided). Recall the 4 steps … • Step 1: only need to specify • Step 2: Test statistic where p0H0: p = p0Haz = (pê à p0)=û0û0= p0(1 à p0)=nq4/5/11 Lecture 18 5 Hypothesis Testing for p (continued) • Step 3: The P-value will be equal to P(Z > z) for 1-sided (upper tail) P(Z < z) for 1-sided (lower tail) 2 P(Z > |z|) for 2-sided • Step 4: Compare the P-value with the significance level and draw your conclusion. Ha: p > p0Ha: p < p0Ha: p6=p0ë4/5/11 Lecture 18 6 “Biased” one-Euro Coin? • A group of statistics students spun the Belgian one-Euro coin 250 times, and heads came up 140 times. • p = P(H) in each spin • Claim: the coin is biased (more specifically, p is greater than 0.5)4/5/11 Lecture 18 7 “Biased” one-Euro Coin? (continued) • Sample: 140 heads among 250 spins of a Belgian one-Euro coin (a hint) • p = P(H) in each spin • H0: p = 0.5 vs Ha: p > 0.5 (one-sided upper) • P-value = P(Z > 1.897) = 1 – 0.9713 = 0.0287 • … Conclude based on a given z = (140=250 à0:5) ä 0:5(1 à0:5)=250p= 1:897ë4/5/11 Lecture 18 8 “Biased” one-Euro Coin? (continued) What about a 95% CI for p ? Note: Margin of error 95% CI = [0.56 – 0.06, 0.56 + 0.06] = [0.5, 0.62] ëm = zãpê(1 àpê)=np= 1:96 0:56(1 à0:56)=250pù 0:064/5/11 Lecture 18 9 Take Home Message • CI for a single proportion p • Margin of error m in CI • Hypothesis testing for p: 4 steps • Assumption: large n How large ? np0õ 10 and n(1 àp0) õ


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UNC-Chapel Hill STOR 155 - Inference for a Single Proportion

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