Three Drivers for Every Van

A logistics company predicts how much freight a depot moves per day from two features:

$x_1$ — number of delivery vans
$x_2$ — number of drivers on the payroll

Five depots make up the training set:

Depot	Vans $x_1$	Drivers $x_2$	Freight $y$ (tonnes/day)
A	$1$	$3$	$7$
B	$2$	$6$	$11$
C	$3$	$9$	$16$
D	$4$	$12$	$22$
E	$5$	$15$	$27$

The model has no bias term: $\hat{y} = w_1x_1 + w_2x_2$ , and $X$ is the $5 \times 2$ design matrix whose columns are $x_1$ and $x_2$ .

The team tries the normal equations $X^\top X\,w = X^\top y$ , finds they cannot be solved, and switches to ridge regression: choose $w$ to minimise

$\lVert y - Xw \rVert^2 + \lambda \lVert w \rVert^2, \qquad \lambda = 50$

whose closed-form solution is

$\hat{w} = (X^\top X + \lambda I)^{-1}X^\top y$

A new depot has just opened, running $6$ vans with $20$ drivers.

How much freight does the ridge model predict for it?

Round your answer to two decimal places.

Three Drivers for Every Van

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