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%TCIDATA{Created=Wednesday, September 30, 1998 11:01:54}
%TCIDATA{LastRevised=Tuesday, January 25, 2005 10:43:05}
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\input{tcilatex}

\begin{document}


\section{Exam}

\section{Text}

\textbf{\Large Fysikalisk kemi, Chalmers}

Enkel linj\"{a}r regression

\subsection{Setup}

Title: Tr\"{a}ningstentamen

Submit:Click to Grade

$\limfunc{nplaces}(x,n)=1.0\left\lfloor 10^{n}x+0.5\right\rfloor /10^{n}$

$\limfunc{rejectarea}:=(\limfunc{outside},\limfunc{inside})$

Points: 10

CSTFile:Help

Choices: Break, Permute

Print Choices: a,b,c,d,e,f,g,h

\section{Question}

\section{Comment}

\subsection{Setup}

$\mu :=0$

$\sigma :=\func{rand}\left( 11,16\right) /10$

$n:=12$

$k:=2$

$\alpha :=\func{rand}(\{0.01,0.025,0.05,0.1\})$

$\beta :=\left( 
\begin{array}{cc}
16 & 13%
\end{array}%
\right) $

$\xi _{1}:=\func{rand}\left( 300,400\right) /100$

$\xi _{2}:=\func{rand}\left( 25,37\right) /10$

$\limfunc{normal}=\func{NormalInv}\left( \func{rand}(1000000)/1000000.0;\mu
,\sigma \right) $

$\func{xvar}=\func{rand}(20,50)/10$

$e:=\left( 
\begin{array}{cccccccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1%
\end{array}%
\right) $

$\epsilon :=\left( 
\begin{array}{cccccccccccc}
\limfunc{normal} & \limfunc{normal} & \limfunc{normal} & \limfunc{normal} & 
\limfunc{normal} & \limfunc{normal} & \limfunc{normal} & \limfunc{normal} & 
\limfunc{normal} & \limfunc{normal} & \limfunc{normal} & \limfunc{normal}%
\end{array}%
\right) ^{T}$

$x:=\left( 
\begin{array}{cccccccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 
\func{xvar} & \func{xvar} & \func{xvar} & \func{xvar} & \func{xvar} & \func{%
xvar} & \func{xvar} & \func{xvar} & \func{xvar} & \func{xvar} & \func{xvar}
& \func{xvar}%
\end{array}%
\right) ^{T}$

$y:=x\beta ^{T}+\epsilon $

$c:=x^{T}x$

$b=\left( x^{T}x\right) ^{-1}x^{T}y$

$m:=\frac{c_{1,2}}{c_{1,1}}$

$v:=\left( \frac{1}{n-k}\left( y-xb\right) ^{T}\left( y-xb\right) \right)
_{1,1}$

$s_{x}:=c_{2,2}-\frac{c_{1,2}^{2}}{c_{1,1}}$

$s_{y}:=\left( y^{T}y-\frac{\left( e^{T}y\right) ^{2}}{c_{1,1}}\right)
_{1,1} $

$t:=\limfunc{nplaces}\left( \func{TInv}(1-\frac{\alpha }{2};n-k),3\right) $

$l_{1}:=\limfunc{nplaces}\left( b_{2,1}-t\sqrt{\frac{v}{s_{x}}},3\right) $

$r_{1}:=\limfunc{nplaces}\left( b_{2,1}+t\sqrt{\frac{v}{s_{x}}},3\right) $

$l_{2}:=b_{1,1}+b_{2,1}\xi _{1}-t\sqrt{v}\sqrt{\frac{1}{n}+\frac{\left( \xi
_{1}-m\right) ^{2}}{s_{x}}}$

$r_{2}:=b_{1,1}+b_{2,1}\xi _{1}+t\sqrt{v}\sqrt{\frac{1}{n}+\frac{\left( \xi
_{1}-m\right) ^{2}}{s_{x}}}$

$\limfunc{svar}:=(\limfunc{nplaces}\left( b_{2,1},2\right) ,\limfunc{nplaces}%
\left( l_{1},2\right) ,\limfunc{nplaces}\left( r_{2},2\right) ,\limfunc{%
nplaces}\left( v\left( 1+\frac{1}{n}+\frac{\left( \xi _{2}-m\right) ^{2}}{%
s_{x}}\right) ,2\right) )$

$z:=\left( 
\begin{array}{cc}
\func{float}\left( x_{1,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{1,1}\right) ,1\right) \\ 
\func{float}\left( x_{2,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{2,1}\right) ,1\right) \\ 
\func{float}\left( x_{3,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{3,1}\right) ,1\right) \\ 
\func{float}\left( x_{4,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{4,1}\right) ,1\right) \\ 
\func{float}\left( x_{5,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{5,1}\right) ,1\right) \\ 
\func{float}\left( x_{6,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{6,1}\right) ,1\right) \\ 
\func{float}\left( x_{7,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{7,1}\right) ,1\right) \\ 
\func{float}\left( x_{8,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{8,1}\right) ,1\right) \\ 
\func{float}\left( x_{9,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{9,1}\right) ,1\right) \\ 
\func{float}\left( x_{10,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{10,1}\right) ,1\right) \\ 
\func{float}\left( x_{11,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{11,1}\right) ,1\right) \\ 
\func{float}\left( x_{12,2}\right) & \limfunc{nplaces}\left( \func{float}%
\left( y_{12,1}\right) ,1\right)%
\end{array}%
\right) $

\subsection{Statement}

Ett st\aa lbolag \"{o}nskar kunna tillverka ett st\aa l med en viss p\aa\ f%
\"{o}rhand given seghetsgrad. Man vet att f\"{o}rekomsten av nickel \"{a}r v%
\"{a}sentlig f\"{o}r st\aa lets seghet men fr\aa gan \"{a}r: Hur p\aa verkar
nickel segheten? F\"{o}r att f\aa\ en f\"{o}rsta id\'{e} om sambandet mellan
st\aa lets seghet och f\"{o}rekomsten av nickel tillverkade man $12$
provlegeringar d\"{a}r man exakt visste hur stor procent nickel som f\"{o}%
rekom och m\"{a}tte p\aa\ dessa st\aa lets seghet. M\"{a}tfelen antages vara
oberoende normalf\"{o}rdelade med v\"{a}ntev\"{a}rdet $\mu $ och
standardavvikelsen $\sigma $. F\"{o}ljande var de data man erh\"{o}ll

\[
\begin{array}{lcccccccccccc}
\text{Nickel} & 
%TCIMACRO{\FORMULA{x_{1,2}}{x_{1,2}}{evaluatenum} }%
%BeginExpansion
x_{1,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{2,2}}{x_{2,2}}{evaluatenum} }%
%BeginExpansion
x_{2,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{3,2}}{x_{3,2}}{evaluatenum} }%
%BeginExpansion
x_{3,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{4,2}}{x_{4,2}}{evaluatenum} }%
%BeginExpansion
x_{4,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{5,2}}{x_{5,2}}{evaluatenum} }%
%BeginExpansion
x_{5,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{6,2}}{x_{6,2}}{evaluatenum} }%
%BeginExpansion
x_{6,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{7,2}}{x_{7,2}}{evaluatenum} }%
%BeginExpansion
x_{7,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{8,2}}{x_{8,2}}{evaluatenum} }%
%BeginExpansion
x_{8,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{9,2}}{x_{9,2}}{evaluatenum} }%
%BeginExpansion
x_{9,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{10,2}}{x_{10,2}}{evaluatenum} }%
%BeginExpansion
x_{10,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{11,2}}{x_{11,2}}{evaluatenum} }%
%BeginExpansion
x_{11,2}
%EndExpansion
& 
%TCIMACRO{\FORMULA{x_{12,2}}{x_{12,2}}{evaluatenum} }%
%BeginExpansion
x_{12,2}
%EndExpansion
\\ 
\text{Seghet} & 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{1},1\right) }{\left( %
%\begin{array}{cc}
%y_{1}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{1}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{2},1\right) }{\left( %
%\begin{array}{cc}
%y_{2}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{2}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{3},1\right) }{\left( %
%\begin{array}{cc}
%y_{3}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{3}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{4},1\right) }{\left( %
%\begin{array}{cc}
%y_{4}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{4}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{5},1\right) }{\left( %
%\begin{array}{cc}
%y_{5}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{5}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{6},1\right) }{\left( %
%\begin{array}{cc}
%y_{6}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{6}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{7},1\right) }{\left( %
%\begin{array}{cc}
%y_{7}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{7}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{8},1\right) }{\left( %
%\begin{array}{cc}
%y_{8}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{8}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{9},1\right) }{\left( %
%\begin{array}{cc}
%y_{9}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{9}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{10},1\right) }{\left( %
%\begin{array}{cc}
%y_{10}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{10}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{11},1\right) }{\left( %
%\begin{array}{cc}
%y_{11}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum} }%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{11}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) 
%EndExpansion
& 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( y_{12},1\right) }{\left( %
%\begin{array}{cc}
%y_{12}\limfunc{nplaces} & \limfunc{nplaces}%
%\end{array}\right) }{evaluatenum}}%
%BeginExpansion
\left( %
\begin{array}{cc}
y_{12}\limfunc{nplaces} & \limfunc{nplaces}%
\end{array}\right) %
%EndExpansion
\end{array}%
\]

I samtliga svar nedan skall alltid anges 2 decimalers noggranhet (t.ex $%
12.00 $).

\subsubsection{Substatement}

Ge l\"{a}mplig figur.

\subsubsection{Substatement}

Skatta en regressionsmodell och ange riktningskoefficienten.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{1}$)

\subsubsection{Substatement}

Ge ett $%
%TCIMACRO{%
%\FORMULA{\left\lceil 100\left( 1-\alpha \right) \right\rceil }{\left\lceil 100-100\alpha \right\rceil }{evaluatenum}}%
%BeginExpansion
\left\lceil 100-100\alpha \right\rceil %
%EndExpansion
$ procentigt symmetriskt konfidensintervall f\"{o}r $\beta $. Svara med
intervallets l\"{a}gsta v\"{a}rde.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{2}$)

\subsubsection{Substatement}

Ge ett $%
%TCIMACRO{%
%\FORMULA{\left\lceil 100\left( 1-\alpha \right) \right\rceil }{\left\lceil 100-100\alpha \right\rceil }{evaluatenum}}%
%BeginExpansion
\left\lceil 100-100\alpha \right\rceil %
%EndExpansion
$ procentigt konfidensintervall f\"{o}r linjen n\"{a}r $x=%
%TCIMACRO{\FORMULA{\xi _{1}}{\xi _{1}}{evaluatenum}}%
%BeginExpansion
\xi _{1}%
%EndExpansion
$. Svara med intervallets h\"{o}gsta v\"{a}rde.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{3}$)

\subsubsection{Substatement}

Ber\"{a}kna vilken seghetsgrad som kan f\"{o}rv\"{a}ntas samt denna
seghetsgrads variation om andelen nickel \"{a}r $%
%TCIMACRO{\FORMULA{\xi _{2}}{\xi _{2}}{evaluatenum}}%
%BeginExpansion
\xi _{2}%
%EndExpansion
$. Ange variationen.

\paragraph{Choices}

InputField(MATH)

GradeProc: givecredit($\limfunc{response}=\limfunc{svar}_{4}$)

\subsection{Solution}

F\"{o}rst ber\"{a}knar vi 
\begin{eqnarray*}
\sum (x_{i}-\bar{x})^{2} &=&%
%TCIMACRO{\FORMULA{c_{2,2}}{c_{2,2}}{evaluate}}%
%BeginExpansion
c_{2,2}%
%EndExpansion
-%
%TCIMACRO{\FORMULA{c_{1,1}}{c_{1,1}}{evaluate}}%
%BeginExpansion
c_{1,1}%
%EndExpansion
\times \left( \frac{%
%TCIMACRO{\FORMULA{c_{1,2}}{c_{1,2}}{evaluatenum}}%
%BeginExpansion
c_{1,2}%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{c_{1,1}}{c_{1,1}}{evaluate}}%
%BeginExpansion
c_{1,1}%
%EndExpansion
}\right) ^{2}=%
%TCIMACRO{\FORMULA{s_{x}}{s_{x}}{evaluatenum} }%
%BeginExpansion
s_{x}
%EndExpansion
\\
\sum (y_{i}-\bar{y})^{2} &=&%
%TCIMACRO{\FORMULA{y^{T}y}{yy^{T}}{evaluate}}%
%BeginExpansion
yy^{T}%
%EndExpansion
-%
%TCIMACRO{\FORMULA{c_{1,1}}{c_{1,1}}{evaluate}}%
%BeginExpansion
c_{1,1}%
%EndExpansion
\times \left( \frac{%
%TCIMACRO{\FORMULA{ey}{ye}{evaluate}}%
%BeginExpansion
ye%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{c_{1,1}}{c_{1,1}}{evaluate}}%
%BeginExpansion
c_{1,1}%
%EndExpansion
}\right) ^{2}=%
%TCIMACRO{\FORMULA{s_{y}}{s_{y}}{evaluate} }%
%BeginExpansion
s_{y}
%EndExpansion
\\
b &=&\frac{%
%TCIMACRO{%
%\FORMULA{\left( yx\right) _{1,2}}{xy\left[ 1,2\right] }{evaluate}}%
%BeginExpansion
xy\left[ 1,2\right] %
%EndExpansion
-%
%TCIMACRO{\FORMULA{c_{1,2}}{c_{1,2}}{evaluate}}%
%BeginExpansion
c_{1,2}%
%EndExpansion
\times \frac{%
%TCIMACRO{\FORMULA{ey}{ye}{evaluate}}%
%BeginExpansion
ye%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{c_{1,1}}{c_{1,1}}{evaluate}}%
%BeginExpansion
c_{1,1}%
%EndExpansion
}}{%
%TCIMACRO{\FORMULA{s_{x}}{s_{x}}{evaluate}}%
%BeginExpansion
s_{x}%
%EndExpansion
}=%
%TCIMACRO{\FORMULA{b_{2,1}}{b_{2,1}}{evaluate} }%
%BeginExpansion
b_{2,1}
%EndExpansion
\\
a &=&\frac{%
%TCIMACRO{\FORMULA{ey}{ye}{evaluate}}%
%BeginExpansion
ye%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{c_{1,1}}{c_{1,1}}{evaluate}}%
%BeginExpansion
c_{1,1}%
%EndExpansion
}-%
%TCIMACRO{\FORMULA{b_{2,1}}{b_{2,1}}{evaluate}}%
%BeginExpansion
b_{2,1}%
%EndExpansion
\frac{%
%TCIMACRO{\FORMULA{c_{1,2}}{c_{1,2}}{evaluate}}%
%BeginExpansion
c_{1,2}%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{c_{1,1}}{c_{1,1}}{evaluate}}%
%BeginExpansion
c_{1,1}%
%EndExpansion
}=%
%TCIMACRO{\FORMULA{b_{1,1}}{b_{1,1}}{evaluate} }%
%BeginExpansion
b_{1,1}
%EndExpansion
\\
s^{2} &=&\frac{1}{%
%TCIMACRO{\FORMULA{c_{1,1}}{c_{1,1}}{evaluate}}%
%BeginExpansion
c_{1,1}%
%EndExpansion
}\left( 
%TCIMACRO{\FORMULA{s_{y}}{s_{y}}{evaluate}}%
%BeginExpansion
s_{y}%
%EndExpansion
-%
%TCIMACRO{\FORMULA{\left( b_{2,1}\right) ^{2}}{b_{2,1}^{2}}{evaluate}}%
%BeginExpansion
b_{2,1}^{2}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{s_{x}}{s_{x}}{evaluate}}%
%BeginExpansion
s_{x}%
%EndExpansion
\right) =%
%TCIMACRO{\FORMULA{v}{v}{evaluate}}%
%BeginExpansion
v%
%EndExpansion
\end{eqnarray*}

\begin{enumerate}
\item Figur h\"{a}r!\FRAME{fhFX}{7.001cm}{5.002cm}{0pt}{}{}{Plot}{\special%
{language "Scientific Word";type "MAPLEPLOT";width 7.001cm;height
5.002cm;depth 0pt;display "USEDEF";plot_snapshots FALSE;mustRecompute
FALSE;lastEngine "MuPAD";xmin "1.5";xmax "5.5";xviewmin "1.5";xviewmax
"5.5";yviewmin "38";yviewmax "85";viewset"XY";rangeset"X";plottype
4;numpoints 100;plotstyle "patchnogrid";axesstyle "normal";xis \TEXUX{z};yis
\TEXUX{y};var1name \TEXUX{$z$};var2name \TEXUX{$y$};function
\TEXUX{$z$};linecolor "black";linestyle 1;pointplot TRUE;pointstyle
"circle";linethickness 1;lineAttributes "Solid";var1range
"1.5,5.5";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle
"Point";}}

\item S\"{o}kt skattad regressionslinje blir%
\[
\hat{Y}=%
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( b_{1,1},2\right) }{\left( %
%\begin{array}{cc}
%\left( \limfunc{nplaces}\right) b_{1,1} & 2.0\limfunc{nplaces}%
%\end{array}\right) }{evaluatenum}}%
%BeginExpansion
\left( %
\begin{array}{cc}
\left( \limfunc{nplaces}\right) b_{1,1} & 2.0\limfunc{nplaces}%
\end{array}\right) %
%EndExpansion
+%
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( b_{2,1},2\right) }{\left( %
%\begin{array}{cc}
%\left( \limfunc{nplaces}\right) b_{2,1} & 2.0\limfunc{nplaces}%
%\end{array}\right) }{evaluatenum}}%
%BeginExpansion
\left( %
\begin{array}{cc}
\left( \limfunc{nplaces}\right) b_{2,1} & 2.0\limfunc{nplaces}%
\end{array}\right) %
%EndExpansion
x 
\]

\item Ett symmetriskt konfidensintervallet f\"{o}r $\beta $ n\"{a}r
standardavvikelsen \"{a}r ok\"{a}nd \"{a}r 
\[
\left( b-t_{\alpha /2}\left( n-2\right) \frac{s}{\sqrt{\sum (x_{i}-\bar{x}%
)^{2}}},a+t_{\alpha /2}\left( n-2\right) \frac{s}{\sqrt{\sum (x_{i}-\bar{x}%
)^{2}}}\right) 
\]%
varf\"{o}r s\"{o}kt konfidensintervall blir 
\[
\left( 
%TCIMACRO{\FORMULA{b_{2,1}}{b_{2,1}}{evaluatenum}}%
%BeginExpansion
b_{2,1}%
%EndExpansion
-%
%TCIMACRO{\FORMULA{t}{t}{evaluatenum}}%
%BeginExpansion
t%
%EndExpansion
\sqrt{\frac{%
%TCIMACRO{\FORMULA{v}{v}{evaluatenum}}%
%BeginExpansion
v%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{s_{x}}{s_{x}}{evaluatenum}}%
%BeginExpansion
s_{x}%
%EndExpansion
}},%
%TCIMACRO{\FORMULA{b_{2,1}}{b_{2,1}}{evaluatenum}}%
%BeginExpansion
b_{2,1}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{t}{t}{evaluatenum}}%
%BeginExpansion
t%
%EndExpansion
\sqrt{\frac{%
%TCIMACRO{\FORMULA{v}{v}{evaluatenum}}%
%BeginExpansion
v%
%EndExpansion
}{%
%TCIMACRO{\FORMULA{s_{x}}{s_{x}}{evaluatenum}}%
%BeginExpansion
s_{x}%
%EndExpansion
}}\right) =\left( 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( l_{1},2\right) }{\left( %
%\begin{array}{cc}
%l_{1}\limfunc{nplaces} & 2.0\limfunc{nplaces}%
%\end{array}\right) }{evaluatenum}}%
%BeginExpansion
\left( %
\begin{array}{cc}
l_{1}\limfunc{nplaces} & 2.0\limfunc{nplaces}%
\end{array}\right) %
%EndExpansion
,%
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( r_{1},2\right) }{\left( %
%\begin{array}{cc}
%r_{1}\limfunc{nplaces} & 2.0\limfunc{nplaces}%
%\end{array}\right) }{evaluatenum}}%
%BeginExpansion
\left( %
\begin{array}{cc}
r_{1}\limfunc{nplaces} & 2.0\limfunc{nplaces}%
\end{array}\right) %
%EndExpansion
\right) \text{.} 
\]

\item Ett $95$ procentigt konfidensintervall f\"{o}r linjen n\"{a}r $\sigma $
ok\"{a}nd \"{a}r%
\[
a+bx\pm t_{\alpha /2}\left( n-2\right) s\sqrt{\frac{1}{n}+\frac{(x-\bar{x}%
)^{2}}{\sum (x_{i}-\bar{x})^{2}}} 
\]%
med data insatt erh\aa lls%
\[
%TCIMACRO{\FORMULA{b_{1,1}}{b_{1,1}}{evaluatenum}}%
%BeginExpansion
b_{1,1}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{b_{2,1}}{b_{2,1}}{evaluatenum}}%
%BeginExpansion
b_{2,1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\xi _{1}}{\xi _{1}}{evaluatenum}}%
%BeginExpansion
\xi _{1}%
%EndExpansion
\pm 
%TCIMACRO{\FORMULA{t}{t}{evaluatenum}}%
%BeginExpansion
t%
%EndExpansion
\sqrt{%
%TCIMACRO{\FORMULA{v}{v}{evaluatenum}}%
%BeginExpansion
v%
%EndExpansion
}\sqrt{\frac{1}{%
%TCIMACRO{\FORMULA{n}{n}{evaluatenum}}%
%BeginExpansion
n%
%EndExpansion
}+\frac{\left( 
%TCIMACRO{\FORMULA{\xi _{1}}{\xi _{1}}{evaluatenum}}%
%BeginExpansion
\xi _{1}%
%EndExpansion
-%
%TCIMACRO{\FORMULA{m}{m}{evaluatenum}}%
%BeginExpansion
m%
%EndExpansion
\right) ^{2}}{%
%TCIMACRO{\FORMULA{s_{x}}{s_{x}}{evaluatenum}}%
%BeginExpansion
s_{x}%
%EndExpansion
}} 
\]%
vilket ger%
\[
\left( 
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( l_{2},2\right) }{\left( %
%\begin{array}{cc}
%l_{2}\limfunc{nplaces} & 2.0\limfunc{nplaces}%
%\end{array}\right) }{evaluatenum}}%
%BeginExpansion
\left( %
\begin{array}{cc}
l_{2}\limfunc{nplaces} & 2.0\limfunc{nplaces}%
\end{array}\right) %
%EndExpansion
,%
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( r_{2},2\right) }{\left( %
%\begin{array}{cc}
%r_{2}\limfunc{nplaces} & 2.0\limfunc{nplaces}%
%\end{array}\right) }{evaluatenum}}%
%BeginExpansion
\left( %
\begin{array}{cc}
r_{2}\limfunc{nplaces} & 2.0\limfunc{nplaces}%
\end{array}\right) %
%EndExpansion
\right) 
\]

\item F\"{o}ljande seghetgrad kan f\"{o}rv\"{a}ntas 
\[
y\left( 
%TCIMACRO{\FORMULA{\xi _{2}}{\xi _{2}}{evaluatenum}}%
%BeginExpansion
\xi _{2}%
%EndExpansion
\right) =%
%TCIMACRO{\FORMULA{b_{1,1}}{b_{1,1}}{evaluatenum}}%
%BeginExpansion
b_{1,1}%
%EndExpansion
+%
%TCIMACRO{\FORMULA{b_{2,1}}{b_{2,1}}{evaluatenum}}%
%BeginExpansion
b_{2,1}%
%EndExpansion
\times 
%TCIMACRO{\FORMULA{\xi _{2}}{\xi _{2}}{evaluatenum}}%
%BeginExpansion
\xi _{2}%
%EndExpansion
=%
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( b_{1,1}+b_{2,1}\xi _{2},2\right) }{%
%\begin{array}{cc}
%\left( \limfunc{nplaces}\right) \left( b_{1,1}+\xi _{2}b_{2,1}\right) & 2.0\limfunc{nplaces}%
%\end{array}\allowbreak }{evaluatenum}}%
%BeginExpansion
%
\begin{array}{cc}
\left( \limfunc{nplaces}\right) \left( b_{1,1}+\xi _{2}b_{2,1}\right) & 2.0\limfunc{nplaces}%
\end{array}\allowbreak %
%EndExpansion
\]%
och dess variation \"{a}r%
\[
%TCIMACRO{\FORMULA{v}{v}{evaluatenum}}%
%BeginExpansion
v%
%EndExpansion
\left( 1+\frac{1}{%
%TCIMACRO{\FORMULA{n}{n}{evaluatenum}}%
%BeginExpansion
n%
%EndExpansion
}+\frac{\left( 
%TCIMACRO{\FORMULA{\xi _{2}}{\xi _{2}}{evaluatenum}}%
%BeginExpansion
\xi _{2}%
%EndExpansion
-%
%TCIMACRO{\FORMULA{m}{m}{evaluatenum}}%
%BeginExpansion
m%
%EndExpansion
\right) ^{2}}{%
%TCIMACRO{\FORMULA{s_{x}}{s_{x}}{evaluatenum}}%
%BeginExpansion
s_{x}%
%EndExpansion
}\right) =%
%TCIMACRO{%
%\FORMULA{\limfunc{nplaces}\left( v\left( 1+\frac{1}{n}+\frac{\left( \xi _{2}-m\right) ^{2}}{s_{x}}\right) ,2\right) }{%
%\begin{array}{cc}
%v\left( \limfunc{nplaces}\right) \left( \frac{1}{n}+\frac{1}{s_{x}}\left( -1.0m+\xi _{2}\right) ^{2}+1.0\right) & 2.0\limfunc{nplaces}%
%\end{array}\allowbreak }{evaluatenum}}%
%BeginExpansion
%
\begin{array}{cc}
v\left( \limfunc{nplaces}\right) \left( \frac{1}{n}+\frac{1}{s_{x}}\left( -1.0m+\xi _{2}\right) ^{2}+1.0\right) & 2.0\limfunc{nplaces}%
\end{array}\allowbreak %
%EndExpansion
\text{.} 
\]
\end{enumerate}

\end{document}