User:Inyuki/MathNotes

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Conditional expectation
$$ \mathbb{E} ( \mathbb{E} (Y|X) ) = \mathbb{E} (Y) $$

Coef
$$b_0 = \overline{y} - b_1 \overline{x}$$

$$b_1 = r \cdot \frac{s_y}{s_x} $$

Interaction
Degrees of freedom: $$(I - 1)(J - 1)$$

SS-Interaction: $$K\sum_{i=1}^I \sum_{j=1}^J ( \overline{y}_{ij\cdot} - \overline{y}_{i\cdot\cdot} - \overline{y}_{\cdot j \cdot} + \overline{y}_{\cdot \cdot \cdot} )^2$$

SS-Within
Degrees of freedom: $$n_{\cdot} - IJ$$

SS-Within: $$\sum_{i=1}^I \sum_{j=1}^J \sum_{k=1}^K ( y_{ijk} - \overline{y}_{ij \cdot})^2$$

S503 HW8
$$s_{\mathrm{pooled}} = \sqrt{\frac{\sum_{i=1}^I (n_i - 1) s_i^2}{n_\cdot - I}} = \sqrt{\frac{ (4-1) 3.559^2 + (5-1) 3.240^2 + (4-1) 3.464^2 }{13 - 3}} = \sqrt{11.59877} \approx 3.406$$