Property 1: If a sample is made as described in Definition 1, with the xij independently and normally distributed and with all σj2 equal, then
where MSj = SSj / dfj and dfj = nj – 1. Thus,
If all the σj2 are equal, the third assertion follows by Property 2 of Chi-square Distribution.
Proof: The assertion about the degrees of freedom is clear. The proof of the assertion about the SS is as follows:
Proof: Since ē = 0 and eij = xij – x̄j, by Definition 1, we have