Power of a poll to detect a lead

Question

A poll of n = 1{,}000 voters tests H0: p = 0.5 against H1: p 0.5, where p is a candidate's support. It declares a real lead (rejects H0) if the number of supporters is 526 or more. Suppose the candidate's true support is p = 0.54. What is the power of this test (the probability it correctly detects the lead)?

Accepted Answer

Model the count under H1. With p = 0.54 and n = 1{,}000: mean = 540, and 	ext{sd} = \sqrt{1000 	imes 0.54 	imes 0.46} = \sqrt{248.4} \approx 15.76. Compute power with continuity correction. "Reject" is X \ge 526, boundary 525.5: 	ext{power} = P(X \ge 525.5 \mid p = 0.54) = 1 - \Phi\!\left(\frac{525.5 - 540}{15.76}ight) = 1 - \Phi(-0.92) = \Phi(0.92) \approx 0.82. So the poll detects a true 4-point lead about 82% of the time. (The false-positive rate under a true tie is P(X \ge 525.5 \mid p = 0.5) \approx 0.053, so the test is properly sized.) Takeaway. Power for a count test is the alte…

Power of a poll to detect a lead

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