A table is given the data: table
Based on this table, we computed
$\sum_{i=1}^{n} X_{i1}^2 = 471$, $\sum_{i=1}^{n} X_{i2}^2 = 163.84$, $\sum_{i=1}^{n} X_{i1}X_{i2} = 235$
$\sum_{i=1}^{n} X_{i1}Y_i = 4915.3$, $\sum_{i=1}^{n} X_{i2}Y_i = 3103.66$
We consider the following model involving both independent variables and an intercept: $$Y_i = \beta_0 + \beta_1X_{i1} + \beta_2X_{i2} + \epsilon_i$$
where $\beta_j, j = 0,1,2$ are $3$ parameters and $\epsilon_i$ are pairwise indepedent random errors with mean $0$ and common variance $\sigma^2$. In the matrix notation, the model is
$$Y = X \beta + \epsilon$$
$$ X= \begin{bmatrix} 1 & 7 & 2.6\\ 1 & 1 & 2.9\\ 1 & 11 & 5.6\\ 1 & 11 & 3.1\\ 1 & 7 & 5.2\\ 1 & 11 & 5.5\\ 1 & 3 & 7.1 \end{bmatrix} $$
$$ Y= \begin{bmatrix} 78.5 \\ 74.3 \\ 104.3 \\ 87.6 \\ 95.9 \\ 109.2 \\ 102.7 \end{bmatrix} $$
The least square estimator of $\hat\beta$ of $\beta$ is
$\hat\beta = (X'X)^{-1}X'Y = \begin{bmatrix} 51.7 \\ 1.5 \\ 6.6 \end{bmatrix} $
question:
(a) write the estimated regression surface and interpret each regression coefficient in the context of the data
How do I do that? I have calculated most of $(X'X)$, $(X'X)^{-1}$, $X'Y$. But not sure how to answer this question.
The regression surface is given by
$$\hat{y}=51.7+1.5x_1+6.6x_2.$$
If you plot this bivariate function (independent variables: $x_1$ and $x_2$; dependent variable: $\hat{y}$) you will get something similar to this surface
The coefficients can be interpreted in the following way.
If $x_1$ and $x_2$ are zero then the output is given by the intercept/bias $51.7$ units, which is the coefficient $\beta_0$. If you only increase $x_1$ by one unit, then the output will increase by $1.5$ units, this is the coefficient $\beta_1$. And if you only increase $x_2$ by one unit, then the output will increase by $6.6$ units, this number is the coefficient $\beta_2$.