Differences

This shows you the differences between two versions of the page.

--- sand_box [2024/05/23 08:57] – hkimscil
+++ sand_box [2024/09/25 07:45] (current) – hkimscil
@@ Line 5: / Line 5: @@
 </WRAP>
+{{tabinclude>page1|Top page,page2|Second page,*page3}}
+<tabbed>
+  * page1|top page
+  * page2
+  * *page3
+</tabbed>
 [{{:r.regressionline3.png}}]
-\begin{eqnarray*}
+\begin{align*}
-&\sum{(Y_i - \hat{Y_i})^2} \\
+& \;\;\;\; \sum{(Y_i - \hat{Y_i})^2} \\
 &= \sum{(Y_i - (a + bX_i))^2}  \;\;\; \because \hat{Y_i} = a + bX_i \\
 &= \text{SSE or SS.residual} \;\;\; \text{(and this should be the least value.)} \\
-\end{eqnarray*}
+\end{align*}
 <WRAP box>
-\begin{eqnarray*}
+\begin{align*}
-\text{for a (constant)} \\
+&\text{for a (constant)} \\ \\
-\\
+&\dfrac{\text{d}}{\text{dv}} \sum{(Y_i - (a + bX_i))^2} \\
-\dfrac{\text{d}}{\text{dv}} \sum{(Y_i - (a + bX_i))^2} & = & \sum \dfrac{\text{d}}{\text{dv}} {(Y_i - (a + bX_i))^2} \\
+&=  \sum \dfrac{\text{d}}{\text{dv}} {(Y_i - (a + bX_i))^2} \\
-& = & \sum{2 (Y_i - (a + bX_i))} * (-1) \;\;\;\; \\
+&= \sum{2 (Y_i - (a + bX_i))} * (-1) \;\;\;\; \\
-& \because & \dfrac{\text{d}}{\text{dv for a}} (Y_i - (a+bX_i)) = -1 \\
+&\because \dfrac{\text{d}}{\text{dv for a}} (Y_i - (a+bX_i)) = -1 \\
-& = & -2 \sum{(Y_i - (a + bX_i))} \\
+& = -2 \sum{(Y_i - (a + bX_i))} \\
 \\
-\text{in order to have the least value, the above should be zero} \\
+&\text{in order to have the least value, the above should be zero} \\
 \\
--2 \sum{(Y_i - (a + bX_i))} & = &  0 \\
+&-2 \sum{(Y_i - (a + bX_i))} = 0 \\
-\sum{(Y_i - (a + bX_i))} & = & 0 \\
+&\sum{(Y_i - (a + bX_i))} = 0 \\
-\sum{Y_i} - \sum{a} - b \sum{X_i} & = & 0 \\
+&\sum{Y_i} - \sum{a} - b \sum{X_i} = 0 \\
-\sum{Y_i} - n*{a} - b \sum{X_i} & = & 0 \\
+&\sum{Y_i} - n*{a} - b \sum{X_i} = 0 \\
-n*{a} & = & \sum{Y_i} - b \sum{X_i} \\
+&n*{a} = \sum{Y_i} - b \sum{X_i} \\
-a & = & \dfrac{\sum{Y_i}}{n} - b \dfrac{\sum{X_i}}{n} \\
+&a = \dfrac{\sum{Y_i}}{n} - b \dfrac{\sum{X_i}}{n} \\
-a & = & \overline{Y} - b \overline{X} \\
+&a = \overline{Y} - b \overline{X} \\
-\end{eqnarray*}
+\end{align*}
 </WRAP>