Three Dial Settings, One Valley

A model predicts a house price from its size using a single dial $w$ :

$\hat{y} = w \cdot x$

Its training data is three houses (size in thousands of sq ft, price in hundreds of thousands):

house	$x$	$y$
1	$1$	$3$
2	$2$	$5$
3	$3$	$8$

The total error is the sum of the misses, each counted as a positive amount:

$E(w) = \sum_{i=1}^{3} \left| w x_i - y_i \right|$

Evaluate $E$ at $w = 2$ , $w = 2.5$ and $w = 3$ . Which setting sits lowest in the valley, and what is the average rate of change of the error as $w$ moves from $2$ to $2.5$ ?

Three Dial Settings, One Valley

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