Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 2 / e

Table of contents
- Solution e

Solution

Under

H_{0}

,

(Z_{1}, Z_{2})

is a bivariate standard normal distribution with correlation

ρ = m_{1} / m_{2}

. Rejection occurs either at stage 1 through

Z_{1} \geq u_{1}

, or at stage 2 through

Z_{2} \geq u_{2}

after continuation

l_{1} < Z_{1} < u_{1}

. Hence the Type I error is

α_{actual} = P_{0} (Z_{1} \geq u_{1}) + P_{0} (l_{1} < Z_{1} < u_{1},, Z_{2} \geq u_{2})

= 1 - Φ (u_{1}) + \int_{l_{1}}^{u_{1}} ϕ (z) [1 - Φ (\frac{u _{2} - ρ z}{1 - ρ ^{2}})] d z .

The final lack-of-benefit boundary

l_{2}

affects acceptance, but not the probability of crossing an efficacy boundary.

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