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A. Juriˇ<strong>si</strong>ć in V. Batagelj: Verjetnostni račun in statistika 323 ✬ ✫ (B) Vzorčna disperzija Imejmo normalno populacijo N(µ, σ). Kako bi določili porazdelitev za vzorčno disperzijo S 2 0 = 1 n ali popravljeno vzorčno disperzijo S 2 = 1 n − 1 Raje izračunamo porazdelitev za statistiko χ 2 = nS2 0 (n − 1)S2 = σ2 σ2 = 1 σ 2 n (Xi − X) 2 i=1 n (Xi − X) 2 ? i=1 n (Xi − X) 2 Univerza v Ljubljani ▲ ❙ ▲ ▲ i=1 ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
A. Juriˇ<strong>si</strong>ć in V. Batagelj: Verjetnostni račun in statistika 324 ✬ Preoblikujemo jo lahko takole: χ 2 = 1 σ 2 = 1 σ 2 . . . Vzorčna disperzija n (Xi − µ + µ − X) 2 = i=1 n i=1 (Xi − µ) 2 + 2 (µ − X) σ2 n i=1 (Xi − µ) + n (µ − X)2 σ2 in, ker je n i=1 (Xi − µ) = n(X − µ) = −n(µ − X), dalje χ 2 = n Xi − µ i=1 σ 2 − 1 n n i=1 2 Xi − µ σ = n i=1 Y 2 i − 1 n 2 Yi , n i=1 kjer so Y1, Y2, . . . , Yn paroma neodvisne standardizirano normalno porazdeljene slučajne spremenljivke, Yi = ✫ Xi − µ . σ Univerza v Ljubljani ▲ ❙ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
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✬ ✫ Ljubljana, 6. oktober 2008
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A. Juriˇsić in V. Batagelj: Verje
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