= Solution
The <Box–Cox transformation> searches over transformations of the positive response,
$$
T_\lambda(y)=\begin{cases}(y^\lambda-1)/\lambda,&\lambda\ne0,\\\log y,&\lambda=0.\end{cases}
$$
The plotted horizontal coordinate is $\lambda$ and the vertical coordinate is its maximized <profile likelihood> on the log scale. The likelihood includes the transformation <Jacobian determinant>; omitting it would compare response scales incorrectly. For the Gaussian regression it is, up to constants,
$$
\ell_p(\lambda)=-\frac n2\log\bigl(\mathrm{RSS}_\lambda/n\bigr)
+(\lambda-1)\sum_i\log y_i.
$$
The maximum occurs around $\lambda=2$. The horizontal 95% line is approximately $\ell_p(\widehat\lambda)-\chi^2_{1,0.95}/2$, and its intersections give an approximate interval roughly $1.2$ to $2.8$, read visually. In particular, $\lambda=1$ is below the line, and $\lambda=0$ is far below it.
\b[Square the response:] a convenient transformation is $Y_i^2$, equivalent up to affine rescaling to $T_2(Y_i)=(Y_i^2-1)/2$. Transform energy, not the sunny-days or latitude predictors. Refit and check the <normal linear model> on the transformed scale; the plot is evidence for a transformation, not proof that the transformed residual assumptions hold. Inference on transformed means cannot be inverted naively to obtain original-scale conditional means.
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