# What is the estimated regression surface

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.

$$\hat{y}=51.7+1.5x_1+6.6x_2.$$
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
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 $$\beta_2$$.