1.I.3H1 . \mathrm{I} . 3 \mathrm{H} \quad

Statistics | Part IB, 2003

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

Yi=α+β(xixˉ)+εi,i=1,,n,Y_{i}=\alpha+\beta\left(x_{i}-\bar{x}\right)+\varepsilon_{i}, \quad i=1, \ldots, n,

where x1,,xnx_{1}, \ldots, x_{n} are given constants, xˉ=n1i=1nxi\bar{x}=n^{-1} \sum_{i=1}^{n} x_{i}, and εi\varepsilon_{i} are independent with Eεi=0,Varεi=σ2,i=1,,n\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, xix_{i} & 1 & 3 & 5 & 10 & 12 \ \hline Cost per unit, yiy_{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, Sxy=i=1n(xixˉ)yi=257.4S_{x y}=\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right) y_{i}=-257.4.]

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