Given: $n=5$, $x=[1,2,3,4,5]$, $y=[3,4,2,5,7]$

Compute $w_0$ (intercept), $w_1$ (slope) for $y = w_0 + w_1 x$ via OLS.

Formulas: $w_1 = \frac{\sum (x_i-\bar x)(y_i-\bar y)}{\sum (x_i-\bar x)^2}$, $w_0 = \bar y - w_1 \bar x$

• $\bar x = (1+2+3+4+5)/5 = 3$
• $\bar y = (3+4+2+5+7)/5 =21/5=4.2$

$x_i$	$y_i$	$x_i-\bar x$	$y_i-\bar y$	$(x_i-\bar x)(y_i-\bar y)$	$(x_i-\bar x)^2$
1	3	-2	-1.2	2.4	4
2	4	-1	-0.2	0.2	1
3	2	0	-2.2	0	0
4	5	1	0.8	0.8	1
5	7	2	2.8	5.6	4
Sum numer $=9.0$, denom $=10$

$w_1 = 9/10 =0.9$

$w_0 =4.2 -0.9*3 =4.2-2.7=1.5$

Model: $\hat y =1.5 +0.9 x$

Prediction at $x=6$: $\hat y =1.5 +0.9*6 =1.5+5.4=6.9 \approx 7$

Residuals check: $\hat y_{train}=[2.4,3.3,4.2,5.1,6.0]$, SSE small.

If using normal equations directly: $w=(X^T X)^{-1}X^T y$ gives same.