The nave height $x$, and the nave length $y$ for 16 Gothic-style cathedrals and 9 Romanesque-style cathedrals, all in England, have been recorded, and the corresponding $R$ output (slightly edited) is given below.

You may assume that $x, y$ are in suitable units, and that "style" has been set up as a factor with levels 1,2 corresponding to Gothic, Romanesque respectively.

(a) Explain carefully, with suitable graph(s) if necessary, the results of this analysis.

(b) Using the general model $Y=X \beta+\epsilon$ (in the conventional notation) explain carefully the theory needed for (a).

[Standard theorems need not be proved.]