Concerns about the increasing friction between some machine parts prompted an investigation of four different types of ball bearings. Five different machines were available and each type of ball bearing was tried in each machine. Given the observations on temperature, coded by subtracting the smallest value,

(a) decompose each observation \(y_{i j}\) as \[\begin{aligned} y_{i j}= & \bar{y}_{\bullet \bullet}+\left(\bar{y}_{i \bullet}-\bar{y}_{\bullet \bullet}\right)+\left(\bar{y}_{\bullet j}-\bar{y}_{\bullet \bullet}\right) \\ & +\left(y_{i j}-\bar{y}_{i_{\bullet}}-\bar{y}_{\bullet j}+\bar{y}_{\bullet}\right) \end{aligned}\]

(b) obtain the sum of squares and the degrees of freedom for each component;

(c) construct the analysis of variance table and test for differences among the bearings using \(\alpha=0.01\).

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Machines 12 3 4 5 Ball bearing 1 Ball bearing 2 Ball bearing 3 10 10 8 7 4 6 10 7 12 5 11 8 9 11 12 10 Ball bearing 4 12 8 8 6 11 13