Suppose that we have $J$ samples and, given the parameters, observation $i$ in sample $j$ is  

$$
y_{i,j}\sim N(\mu_{j},\ \tau^{-1})
$$  

for $i=1,\dots,n_{j}$ and $j=1,\dots,J$ .  

Let $\underline{{\boldsymbol{\mu}}}=(\mu_{1},\ldots,\mu_{J})^{T}$ , let $\bar{\underline{{y}}}=(\bar{y}_{1},\dots,\bar{y}_{J})^{T}$ , and let  

$$
S=\sum_{j=1}^{J}\sum_{i=1}^{n_{j}}(y_{i,j}-\Bar{y}_{j})^{2},
$$  

where  

$$
{\bar{y}}_{j}={\frac{1}{n_{j}}}\sum_{i=1}^{n_{j}}y_{i,j}.
$$  

Show that $\underline{y}$ and $S$ are sufficient for $\underline{{\boldsymbol\mu}}$ and $\tau$ . 