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App. stats CW 1 WIP

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henrydatei 2019-02-24 15:36:47 +00:00
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Erasmus/Applied statistics/Coursework 1.pdf
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Erasmus/Applied statistics/Coursework 1.tex
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@ -20,8 +20,18 @@
\section{Task 1}
\subsection{Part (1)}
In the given data were two out of 26 data points with an Al/Be ratio of more than 4.5. That means
\begin{align}
\hat{p} = \frac{2}{26} = \frac{1}{13}\notag
\end{align}
\subsection{Part (2)}
Using the following formula from the lecture we get the 95\% confidence interval:
\begin{align}
\hat{p}&\pm 2\cdot\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \notag \\
\frac{1}{13} &\pm \underbrace{2\cdot\sqrt{\frac{\frac{1}{13}\cdot\frac{12}{13}}{26}}}_{0.1045} \notag
\end{align}
Our 95\% confidence interval is [-0.0276,0.1814] which means that we are 95\% sure that the true proportion lies between -0.0276 and 0.1814.
\subsection{Part (3)}
@ -30,8 +40,77 @@
\pagebreak
\section{Task 2}
\subsection{Part (1)}
\begin{lstlisting}
x = [-4.5, -1, -0.5, -0.15, 0, 0.01, 0.02, 0.05, ...
0.15, 0.2, 0.5, 0.5, 1, 2, 3];
m = mean(x);
s = std(x);
\end{lstlisting}
\begin{center}
\begin{tabular}{p{4cm}|p{7cm}}
null hypothesis & $H_0$: $\mu = 0$ \\
\hline
alternative hypothesis & $H_A$: $\mu\neq 0$ \\
\hline
t-test for $\mu$ & $t=\frac{m-0}{\frac{s}{\sqrt{15}}} =\frac{0.0853}{\frac{1.6031}{\sqrt{15}}} = 0.2062$ \\
\hline
rejection region & \texttt{tinv(0.05,15)} = -1.7531 \\
\hline
conclusion & $t$ lies not in the rejection region so $H_0$ is accepted at the 10\% significance level.
\end{tabular}
\end{center}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
xmin=-3, xmax=3, xlabel=$x$,
ymin=0, ymax=1, ylabel=$y$,
samples=400,
axis y line=middle,
axis x line=middle,
domain=-3:3,
restrict y to domain=0:1,
width = 16cm,
height = 8cm,
]
\addplot[name path=f,blue] {116640000000*sqrt(15)/(143*pi*(x^2+15)^(8))};
\path[name path=axis] (axis cs:1.7531,0) -- (axis cs:3,0);
\path[name path=axis2] (axis cs:-3,0) -- (axis cs:-1.7531,0);
\draw (axis cs:1.7531,0) -- (axis cs:1.7531,1);
\draw (axis cs:-1.7531,0) -- (axis cs:-1.7531,1);
\draw [dotted] (axis cs:0.2062,0) -- (axis cs:0.2062,0.6);
\node at (axis cs:1,0.5) (a) {0.05};
\node at (axis cs:-1,0.5) (a2) {0.05};
\draw (axis cs:1, 0.46) -- (axis cs: 2.2,0.02);
\draw (axis cs:-1, 0.46) -- (axis cs: -2.2,0.02);
\node at (axis cs: 2.0,0.95) (b) {1.7531};
\node at (axis cs: -2.0,0.95) (b2) {-1.7531};
\node at (axis cs: 0.45,0.55) (c) {0.2062};
\node[red] at (axis cs: 2.4,0.78) (d) {rejection region};
\node[red] at (axis cs: -2.4,0.78) (d2) {rejection region};
\begin{scope}[transparency group]
\begin{scope}[blend mode=multiply]
\addplot [thick,color=blue,fill=blue,fill opacity=0.3] fill between[of=f and axis,soft clip={domain=1.7531:3},];
\addplot [thick,color=blue,fill=blue,fill opacity=0.3] fill between[of=f and axis2,soft clip={domain=-3:-1.7531},];
\draw[red,fill=red,opacity=0.2] (axis cs: 1.7531,0) -- (axis cs: 1.7531,1) -- (axis cs: 3,1) -- (axis cs: 3,0) -- (axis cs: 1.7531,0);
\draw[red,fill=red,opacity=0.2] (axis cs: -1.7531,0) -- (axis cs: -1.7531,1) -- (axis cs: -3,1) -- (axis cs: -3,0) -- (axis cs: -1.7531,0);
\end{scope}
\end{scope}
\end{axis}
\end{tikzpicture}
\end{center}
\subsection{Part (2)}
If we reduce the significance level our rejection region gets smaller. With $\alpha = 0.05$ the rejection region will start at \texttt{tinv(0.025,15)} = -2.1314. The $t$ calculated in part (1) won't change $\Rightarrow$ our decision won't change too.
To get the type 2 error we use the MATLAB function \texttt{sampsizepwr} and $type\, 2\, error = 1-power$.
\begin{lstlisting}
testtype = 't';
p0 = [0 1.6031];
p1 = 0.0853;
n = 15;
power = sampsizepwr(testtype,p0,p1,[],n)
\end{lstlisting}
This gives $power = 0.0542\Rightarrow type\, 2\, error = 0.9458$. This is the probability of wrongly accepting $H_0$ when it is false.
\subsection{Part (3)}

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Erasmus/Applied statistics/mathscript.cls
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@ -136,12 +136,12 @@
\definecolor{mygray}{rgb}{0.5,0.5,0.5}
\definecolor{mygreen}{rgb}{0,0.8,0.26}
\lstset{language=[95]Fortran,
\lstset{language=Matlab,
basicstyle=\ttfamily,
keywordstyle=\color{lila},
commentstyle=\color{lightgray},
morecomment=[l]{!\ },% Comment only with space after !
morekeywords={compute, solve, choose, break, switch},
morekeywords={compute, solve, choose, break, switch, sampsizepwr},
stringstyle=\color{mygreen}\ttfamily,
backgroundcolor=\color{white},
showstringspaces=false,