hw5.lyx

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\begin_body

\begin_layout Standard
John Hancock
\end_layout

\begin_layout Standard
CEN 6405
\end_layout

\begin_layout Standard
Homework 5
\end_layout

\begin_layout Standard
May 23nd, 2014
\end_layout

\begin_layout Part*
Homework 5
\end_layout

\begin_layout Section*
17.1
\end_layout

\begin_layout Subsection*
a.
 Quantify main effects and all interactions
\end_layout

\begin_layout Standard
We follow the example 17.3.
 We construct a sign table to quantify the effects and interactions:
\end_layout

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Total/8
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\begin_layout Standard
We read the bottom row of the table to quantify the effects of 
\begin_inset Formula $A$
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, 
\begin_inset Formula $B$
\end_inset

, and 
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.
 They are 
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\end_inset

, 
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\end_inset

, and 
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, respectively.
 We do the same to quantify the two factor interactions.
 The three two factor interactions are 
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 , 
\begin_inset Formula $q_{AC}=1.875$
\end_inset

 and 
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.
 Finally, the three factor interaction is 
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\begin_layout Subsection*
b.
 Quantify percentages of variation explained
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\begin_layout Standard
Again, we follow the example 17.3.
 In order to quantify the percentages of variation explained, we must first
 calculate the total sum of squares (SST).
 The SST is computed as: 
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SST=2^{3}\left(q_{A}^{2}+q_{B}^{2}+q_{C}^{2}+q_{AB}^{2}+q_{AC}^{2}+q_{BC}^{2}+q_{ABC}^{2}\right)\]

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\begin_inset Formula \[
=8\left(1.875^{2}+-13.125^{2}+-21.875^{2}+-1.875^{2}+1.875^{2}+26.875^{2}+8.125^{2}\right)\]

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\end_layout

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=11596.875\]

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\begin_layout Standard
Our text states, 
\begin_inset Quotes eld
\end_inset

The portion of variation explained by the various factors and interactions
 are proportional to the square of the effects.
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\end_inset

 Therefore these portions are: 
\end_layout

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<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{8\times q_{A}^{2}}{SST}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{28.125}{11596.875}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\approx0.24\%$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{8\times q_{B}^{2}}{SST}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{1378.125}{11596.875}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\approx11.88\%$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{8\times q_{C}^{2}}{SST}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{3828.125}{11596.875}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\approx33.01\%$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{8\times q_{AB}^{2}}{SST}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{28.125}{11596.875}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\approx0.24\%$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{8\times q_{AC}^{2}}{SST}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{28.125}{11596.875}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\approx0.24\%$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{8\times q_{BC}^{2}}{SST}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{5778.125}{11596.875}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\approx49.82\%$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{8\times q_{ABC}^{2}}{SST}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\frac{528.125}{11596.875}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\approx4.55\%$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
The rightmost column in the table above quantifies the percentages of the
 variation explained.
 The leftmost column tells us the effect or interaction that explains the
 portion of the variation.
\end_layout

\begin_layout Subsection*
c.
 Sort the variables in the order of decreasing importance.
 
\end_layout

\begin_layout Standard
The variable that explains the largest portion of variation is the most
 important, the variable that explains the second largest portion of variation
 is the second most important, and so on.
 Referring to the table above, we see that the list of variables sorted
 in order of importance is: 
\begin_inset Formula $C,$
\end_inset

 
\begin_inset Formula $B,$
\end_inset

 
\begin_inset Formula $A$
\end_inset

 .
\end_layout

\end_body
\end_document