Solution (source code)

= Solution

At fraction $0.8$, the <Lasso regularization path> lies between the vertical lines marked steps 4 and 5. In that interval the first five predictors have nonzero slopes: $X_1,X_2,X_4,X_5$ are positive and $X_3$ is negative. The sixth predictor remains at zero until the later part of the path, beyond this fraction. Therefore the chosen <Lasso> model includes
$$
\boxed{X_1,X_2,X_3,X_4,X_5\text{, with }X_6\text{ excluded}.}
$$
The <regression intercept> is retained. In particular the negative $X_3$ trace has already left zero at fraction $0.8$; a small slope is not the same as a zero slope. The selected fraction came from ten-fold <K-fold cross-validation>, and is not a numerical value of the ridge penalty in the preceding parts.