Linear regression - find $b$, $s^2$, $R^2$ Suppose you want to fit the model $Y=\alpha+\beta X+\epsilon$ but you don't have the full data set $\left[\begin{matrix}y&X\\\end{matrix}\right]=C$. Instead you only have:
$$C'C=\begin{bmatrix}
100 & 10 & 25\\
10 & 20 & 0\\
25 & 0 & 75
\end{bmatrix}$$
Calculate b, $s^2$ and $R^2$.

My main question is: what is $C'C$?
 I'm new to this topic and I'm used to see this form (in matrix)
$$(X'X)b=X'y$$ and, from this equation, I can compute easily $b$.
Could you give me some hints on how to start to solve this exercise?
 A: Hints: The $C$ matrix refers to having $y$ as its first column and $X$ stacked next to this. Note that $C = \begin{bmatrix}y & X\end{bmatrix}$, so $C' = \begin{bmatrix} y' \\ X'\end{bmatrix}$ and
$$\begin{align*}
C'C &= \begin{bmatrix} y' \\ X'\end{bmatrix} \begin{bmatrix}y & X\end{bmatrix} \\
&= \begin{bmatrix}   \underbrace{y'y}_{1\times 1} & \underbrace{y'X}_{1\times p} \\ \underbrace{X'y}_{p\times 1}& \underbrace{X'X}_{p\times p}\end{bmatrix}.
\end{align*}$$
(We effectively used block matrix multiplication.)
Note that $p=2$ for this problem (simple linear regression). Thus you can read off useful information from parts of the $C'C$ matrix. For example the lower right $p\times p$ ($2\times 2$) submatrix of $C'C$ is $X'X$. There is now enough information to deduce everything we need. Can you go from here? Also note that you can figure out things like $n,\bar{x}$ and other useful quantities from looking at the entries of $X'X$, $y'y$ and $X'y$. Use the definition of $X$, $y$, and matrix multiplication to do this where needed.
