Assume a linear relationship for a company that has several shops is not known. Let $Y_i$ be the profit the shop number $i$ makes in the coming year. Let $x_i$ be the size of the shop number $i$. We assume that for all these shops the following relationship holds. $$Y_i = \alpha + \beta x_i + \epsilon_i$$ where $\epsilon_i$ is a random term for which $E[\epsilon_i] = 0$ and such that $\epsilon_1,\epsilon_2,...$ are $i.i.d$.
$\alpha, \beta$ and $\sigma$ are unknown and must be estimated.
a.) If we want to open a shop with size 3, what is the expected profit in terms of $\alpha$ and $\beta$?
b.) Estimate $\alpha$ and $\beta$ using linear regression for the following data $x_i = 1,2,3,4,5,6,7,8,9$ corresponding to $y_i = 0,1,1,2,1,3,3,3,4$.
For part a, is the expected profit going to be $Y_i = \alpha + \beta x_i$, where $x_i = 3$, since $\epsilon_i$ has mean $0$?
For part b, we have not yet been taught linear regression but I did google it and I know that I must find $a$ and $b$ minimizing the sum of the distance square: $$\sum^n_i (Y_i -(a - bx_i))^2$$ then the estimate of $\alpha$ and $\beta$ are then $a$ and $b$ which minimize the above expression. Also, why would the given data values be necessary to find $\alpha$ and $\beta$? Isn't those values only necessary to find the standard deviation?