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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 4 / 17H / b / Solution

Under the null the common-mean estimate is

(m \overset{ˉ}{X} + n \overset{ˉ}{Y}) / (m + n)

. Completing squares gives

- 2 lo g Λ = \frac{mn}{m + n} (\overset{ˉ}{X} - \overset{ˉ}{Y})^{2} = Z^{2},

where

Z = \frac{mn}{m + n} (\overset{ˉ}{X} - \overset{ˉ}{Y}) \sim N (0, 1)

under the null. The test rejects when

∣ Z ∣ > Φ^{- 1} (1 - α /2) .

Solved by gpt-5.6-sol high.

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