the positive constant is called the 1 100 condent

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Unformatted text preview: to be finite. When should we stop increasing n? Till there is enough guarantee that Yn must be sufficiently close to p ! M. Chen (IE@CUHK) ENGG2430C lecture 10 20 / 29 ATV poll problem revisit: Confidence interval * Question: Given positive constants 0 < δ < 1 and ε , how large should n be, to make sure with at most probability δ , Yn ∈ (p − ε , p + ε )? / That is, find the smallest n so that P (Yn ∈ (p − ε , p + ε )) = P (|Yn − p | ≥ ε ) ≤ δ . / The positive constant ε is called the (1 − δ ) × 100%-confident interval. Exercises: Prove that for any p ∈ [0, 1], p (1 − p ) ≤ 1 . 4 Compute a suitable n for δ = 0.01 and ε = 0.05. (Hint: use Chebyshev inequality.) M. Chen (IE@CUHK) ENGG2430C lecture 10 21 / 29 ATV poll problem revisit: Confidence interval 1 Solution: By Chebyshev inequality and p (1 − p ) ≤ 4 , we obtain an upper bound for P (|Yn − p | ≥ ε ) as P (|Yn − p | ≥ ε ) ≤ = Var(Yn ) ε2 p (1 − p ) 1 ≤ 2 nε 4nε 2 A sufficient condition for P (|Yn − p | ≥ ε ) ≤ δ i...
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