Question

(a) Calculate S using the data matrix Y as in (3.29). (b) Obtain R by calculating ri2, ?"i3, and r23, as in (3.34) and (3.35). (c) Find R using (3.37).

3.10 Use the calcium data in Table 3.4: (a) Calculate S using the data matrix Y as in (3.29) (b) Obtain R by calculating r12,T13, and r23, as in (3.34) and (3.35). (c) Find R using (3.37).

Answer 1

$(a)~Variance-Covariance~matrix=S=\begin{pmatrix} s_{11} & s_{12} &s_{13} \\ s_{21} &s_{22} &s_{23} \\ s_{31} &s_{32} & s_{33} \end{pmatrix}\\ where,~s_{ij}=\frac{1}{10}\sum_{k=1}^{10}(y_{ik}-\bar{y}_i)(y_{jk}-\bar{y}_j),~i,j=1(1)10,~s_{ij}=s_{ji}\\ \bar{y}_i=\frac{1}{10}\sum_{k=1}^{10}y_{ik}$

$Hence,~S=\begin{pmatrix} 140.54444 &49.68000 &1.94122 \\ 49.68000 &72.24844 &3.67609 \\ 1.94122 &3.67609 &0.25012 \end{pmatrix}\\$

$(b)~r_{ij}=\frac{s_{ij}}{\sqrt{s_{ii}s_{jj}}},~i\neq j\\ r_{12}=0.148,~r_{13}=0.356,~r_{23}=0.001\\(c)~R^{3\times 3}=(r_{ij})\\ R=\begin{pmatrix} 1 & 0.148& 0.356\\ 0.148 & 1 & 0.001 \\ 0.356& 0.001 &1 \end{pmatrix}$

Variable Mean
y_1 28.10
y_2 7.18
y_3 3.089