Discussion
If \(S_{1}, S_{2}, \ldots, S_{5}\) are the corrected sum of squares and sum of products matrices based on five independent samples of sizes 100 each coming from a four variate normal population. The distribution of \(\sum_{i=1}^{5} Si\).
Answer: B
\(\sum S_{i}\) is Wishart with \(k = 495\) degrees of freedom.
For a sample of size 100 from a \(p\)-variate normal, the corrected sum of squares and products matrix is \(S_{i} \sim W_{4}(99, \Sigma)\), losing one degree of freedom for the estimated mean. The five samples are independent, and sums of independent Wisharts with the same \(\Sigma\) are Wishart with added degrees of freedom:
\[ k = 5 \times 99 = 495 \]
496 and 499 subtract too few degrees of freedom; 499 treats the five samples as one sample of 500 with a single mean, which they are not.
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