Question

The following data resulted from 5 experimental runa made on four indrpendent variables and a single response y

Answer 1

Since given data is incomplete, we assume it as follows:

The following data resulted from 5 experimental runs made on four independent variables and a single response y.

Here n=5. Write 5 equations with 4 explanatory variables in matrix notation.

Solution:

Here n=5, k=4.

Assuming that the model is

 $y=\beta_{0}+\beta_{1} X_{1}+\beta_{2} X_{2}+\ldots+\beta_{4} X_{4}+\varepsilon$  

the n-tuples of observations are also assumed to follow the same model. Thus they satisfy

$\begin{aligned} &y_{1}=\beta_{0}+\beta_{1} x_{11}+\beta_{2} x_{12}+\ldots+\beta_{4} x_{1 4}+\varepsilon_{1} \\ &\vdots \\ &y_{4}=\beta_{0}+\beta_{1} x_{5 1}+\beta_{2} x_{5 2}+\ldots+\beta_{4} x_{54}+\varepsilon_{5} \end{aligned}$

These 5 equations can be written in matrix notation as

$\left(\begin{array}{l}y_{1} \\ y_{2} \\ \vdots \\ y_{5}\end{array}\right)=\left(\begin{array}{ccccc}1 & x_{11} & x_{12} & \cdots & x_{1 4} \\ 1 & x_{21} & x_{22} & \cdots & x_{2 4} \\ \vdots & \vdots & \vdots & \vdots \\ 1 & x_{5 1} & x_{5 2} & \cdots & x_{54}\end{array}\right)\left(\begin{array}{l}\beta_{0} \\ \beta_{1} \\ \vdots \\ \beta_{4}\end{array}\right)+\left(\begin{array}{l}\varepsilon_{1} \\ \varepsilon_{2} \\ \vdots \\ \varepsilon_{5}\end{array}\right)$