I have a question from my assignment.
For a total of 150 households selected randomly from 3 cities -- A, B and C -- of 50 each, the sample mean and covariance matrices are as below:
For A,
Mean = $\begin{pmatrix} 90\ 40\\ \end{pmatrix}$'
S = $\begin{pmatrix} 25 & 0\\ 0 & 25 \\ \end{pmatrix}$
For B,
Mean = $\begin{pmatrix} 100\ 50\\ \end{pmatrix}$'
S = $\begin{pmatrix} 16 & 9\\ 9 & 9 \\ \end{pmatrix}$
For C,
Mean = $\begin{pmatrix} 80\ 60\\ \end{pmatrix}$'
S = $\begin{pmatrix} 19 & 1\\ 1 & 16 \\ \end{pmatrix}$
Based on this data, is there any significant evidence indicating the residents of the 3 cities have different household incomes and expenses? Assume $$ -( n-1- (p+a)/ 2 ) log ⋏ \sim ^2_{p(a-1)} $$
The problem is I have no idea what "a" and "⋏" stands for. I presume ⋏ is diag( eigenvalues of the matrices).
My idea is to combine all the Mean sample into one whole and Covariance sample into another whole. There after finding $H_0$ using the distribution assumption given. However, what is my "a" and my "⋏"?
Is my thinking correct or do I have to compare A vs B, B vs C and C vs A?