An experimenter wants to compare the time to failure \(y\) after rebuilding a robotic welder by three different methods but adjusting for the covariate \(x=\) age of robotic welder.

Suppose the data, in thousands of hours, are

The resulting sum of squares error are \(S S E_{x}=15.917, S S E_{t r}=10.000\), and \(S S E_{t r, x}=1.833\). Assuming that the analysis of covariance model is reasonable, conduct the \(F\)-test for treatment differences and the \(F\)-test for the covariate \(x\). Take \(\alpha=0.05\).

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Method 3 x y 6 Method 2 y 4 54 534 +53 684 Method I y x x 7 2 5 52 11 6 2 4