Question #40986

Let $M = \begin{pmatrix} \bar{x}_1 & n_1 \\ n_2 & \bar{x}_2 \end{pmatrix}$ be the parameter matrix for two subgroups $G_1$ and $G_2$, where $n_i$ and $\bar{x}_i$ represent the number of observations and the arithmetic mean of subgroup $G_i$ ($i=1,2$), respectively. Consider the matrices $A = \begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$. If the matrix transformation $T(M) = B M A$ yields $T(M) = \begin{pmatrix} 255 & 445 \\ 180 & 315 \end{pmatrix}$, find the combined arithmetic mean of the dataset formed by merging $G_1$ and $G_2$.