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\[E(SS_{W})=I(J-1)\sigma^{2}\] (Bevis (\[E(SS_{W})=\sum_{i=1}^{I}\sum…
\[E(SS_{W})=I(J-1)\sigma^{2}\]
Bevis
\[E(SS_{W})=\sum_{i=1}^{I}\sum_{j=1}^{J}E\left((Y_{ij}-\overline{Y}_{i.})^{2}\right)\]
\[=\sum_{i=1}^{I}\sum_{j=1}^{J}\left((\underbrace{(\mu+\alpha_{i})}_{\mu_{i}}-\underbrace{(\mu+\alpha_{i})}_{\overline{\mu}})^{2}+\underbrace{\frac{J-1}{J}\sigma^{2}}_{\frac{n-1}{n}\sigma^{2}}\right)\]
\[=\sum_{i=1}^{I}\sum_{j=1}^{J}\frac{J-1}{J}\sigma^{2}=IJ\frac{J-1}{J}\sigma^{2}=I(J-1)\sigma^{2}\]
\(Y_{ij}\) tar rollen som \(X_i\)
\(\overline{Y}_i\) tar rollen som \(\overline{X}\)
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