Derive the least squares estimators $\hat{\alpha}$ and $\hat{\beta}$ for the coefficients of the simple linear regression model

$$Y_{i}=\alpha+\beta\left(x_{i}-\bar{x}\right)+\varepsilon_{i}, \quad i=1, \ldots, n,$$

where $x_{1}, \ldots, x_{n}$ are given constants, $\bar{x}=n^{-1} \sum_{i=1}^{n} x_{i}$, and $\varepsilon_{i}$ are independent with $\mathrm{E} \varepsilon_{i}=0, \operatorname{Var} \varepsilon_{i}=\sigma^{2}, i=1, \ldots, n$.

A manufacturer of optical equipment has the following data on the unit cost (in pounds) of certain custom-made lenses and the number of units made in each order:

\begin{tabular}{l|ccccc} No. of units, $x_{i}$ & 1 & 3 & 5 & 10 & 12 \ \hline Cost per unit, $y_{i}$ & 58 & 55 & 40 & 37 & 22 \end{tabular}

Assuming that the conditions underlying simple linear regression analysis are met, estimate the regression coefficients and use the estimated regression equation to predict the unit cost in an order for 8 of these lenses.

[Hint: for the data above, $S_{x y}=\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right) y_{i}=-257.4$.]