Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 3 / e / Solution

With equal allocation and true alternative rates

λ_{T}, λ_{C}

, the asymptotic variance is

V_{A} = \frac{2}{N} (\frac{1}{λ _{T}} + \frac{1}{λ _{C}}) .

The normal approximation requires

∣ lo g R R_{A} ∣/ V_{A} \geq z_{1 - α /2} + z_{1 - β}

. Hence

N = \frac{2 ( z _{1 - α /2} + z _{1 - β} ) ^{2}}{( l o g R R _{A} ) ^{2}} (\frac{1}{λ _{T}} + \frac{1}{λ _{C}}),

rounded upward. Writing

λ_{T} = R R_{A} λ_{C}

expresses it using a specified control rate and the target rate ratio.

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