 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-4/17h/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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) .

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