Residuals Math Example 2

Follow the full solution, then compare it with the other examples linked below.

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Five observed and predicted values:

(y, \hat{y})

(10,8), (15,14), (12,13), (20,19), (8,11)

. Calculate all residuals, verify they sum to 0, and compute

\sum e_i^2

1
Residuals: $e_1=10-8=2$ ; $e_2=15-14=1$ ; $e_3=12-13=-1$ ; $e_4=20-19=1$ ; $e_5=8-11=-3$
2
Sum of residuals: $2+1+(-1)+1+(-3) = 0$ ✓ (confirms a valid LSRL)
3
$\sum e_i^2 = 4+1+1+1+9 = 16$
4
The LSRL minimizes this sum of 16 — any other line would produce a larger $\sum e_i^2$

Answer

Residuals:

2,1,-1,1,-3

; sum=0;

\sum e_i^2=16

The sum of residuals from the LSRL always equals zero (because the LSRL passes through

(\bar{x},\bar{y})

). The sum of squared residuals (SSE) is what the LSRL minimizes — this is the defining property of the least-squares method.

About Residuals

The difference between an observed value and its predicted value from a regression model: $\text{residual} = y - \hat{y}$ (observed minus predicted).

Given [formula], and observed point [formula]: calculate the residual and interpret whether the mode

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