Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
\end{warning}
-\pgfmathdeclarefunction{gauss}{2}{%
- \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
-}
- \pgfplotsset{every axis/.append style={
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- }} \pgfkeys{/pgf/decoration/.cd,
- distance/.initial=10pt
-} \pgfdeclaredecoration{add dim}{final}{
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-\tikzset{dim/.style args={#1,#2}{decoration={add dim,distance=#2},
- decorate,
- postaction={decorate,decoration={text along path,
- raise=#2,
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- text={#1}}}}}
- \begin{figure*}[hb]
- \centering
- {\begin{center} \begin{tikzpicture}
- \pgfplotsset{set layers, axis x line=middle, axis y line=middle}
-\begin{axis}[every axis plot post/.append style={
- mark=none,domain=-3:3,samples=50,smooth},
- axis x line=bottom,
- axis y line=left,
- enlargelimits=upper,
- x=\textwidth/10,
- ytick={0.55},
- yticklabels={\(\frac{1}{\sigma \sqrt{2\pi}}\)},
- xtick={-2,-1,0,1,2},
- x tick label style = {font=\footnotesize},
- xticklabels={\((\mu-2\sigma)\), \((\mu-\sigma)\), \(\mu\), \((\mu+\sigma)\), \((\mu+2\sigma)\)},
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-\fill[red!30] (-3,0) -- plot[id=f3,domain=-3:3,samples=50]
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- \fill[lightgray!30] (-2,0) -- plot[id=f3,domain=-2:2,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (2,0) -- cycle;
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-\end{axis}
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- hide y axis,
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- xlabel={\(Z\)},
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- \addplot {gauss(0,0.75)};
-\end{axis}
-\end{tikzpicture}\end{center}}
- \end{figure*}
+ \begin{figure*}[hb]
+ \centering
+ \include{normal-dist-graph}
+ \end{figure*}
\subsection*{Central limit theorem}
\subsection*{\(p\)-value}
- Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true.
- % table of p-values for strength of evidence
+ \begin{align*}
+ p &= \Pr(\overline{X} \lessgtr \mu(H_1)) \\
+ &= 2 \cdot \Pr(\overline{X} <> \mu(H_1) | \mu = 8)
+ \end{align*}
- \subsection*{Distribution of sample mean}
+ Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true.
- If \(X \sim \operatorname{N}(\mu, \sigma)\), then distribution of sample mean \(\overline{X}\) is also normal with \(\overline{X} \sim \operatorname{N}(\mu, \frac{\sigma}{\sqrt{n}}\).
+ \vspace{0.5em}
+ \begin{tabularx}{23em}{|l|X|}
+ \hline
+ \rowcolor{cas}
+ \(\boldsymbol{p}\) & \textbf{Conclusion} \\
+ \hline
+ \(> 0.05\) & insufficient evidence against \(H_0\) \\
+ \(< 0.05\) (5\%) & good evidence against \(H_0\) \\
+ \(< 0.01\) (1\%) & strong evidence against \(H_0\) \\
+ \(< 0.001\) (0.1\%) & very strong evidence against \(H_0\) \\
+ \hline
+ \end{tabularx}
\subsection*{Statistical significance}
\end{description}
\end{cas}
+ \subsection*{One-tail and two-tail tests}
+
+ \subsubsection*{One tail}
+
+ \begin{itemize}
+ \item \(\mu\) has changed in one direction
+ \item State ``\(H_1: \mu \lessgtr \) known population mean''
+ \end{itemize}
+
+ \subsubsection*{Two tail}
+
+ \begin{itemize}
+ \item Direction of \(\Delta \mu\) is ambiguous
+ \item State ``\(H_1: \mu \ne\) known population mean''
+ \end{itemize}
+
+ For two tail tests:
+ \begin{align*}
+ p\text{-value} &= \Pr(|\overline{X} - \mu| \ge |\overline{x}_0 - \mu|) \\
+ &= \left( |Z| \ge \left|\dfrac{\overline{x}_0 - \mu}{\sigma \div \sqrt{n}} \right| \right)
+ \end{align*}
+
+ \subsection*{Modulus notation for two tail}
+
+ \(\Pr(|\overline{X} - \mu| \ge a) \implies\) ``the probability that the distance between \(\overline{\mu}\) and \(\mu\) is \(\ge a\)''
+
+ \subsection*{Inverse normal}
+
+ \begin{cas}
+ \verb;invNormCdf("L", ;\(\alpha\)\verb;, ;\(\dfrac{\sigma}{n^\alpha}\)\verb;, ;\(\mu\)\verb;);
+ \end{cas}
+
+ \subsection*{Errors}
+
+ \begin{description}[labelwidth=2.5cm, labelindent=0.5cm]
+ \item [Type I error] \(H_0\) is rejected when it is \textbf{true}
+ \item [Type II error] \(H_0\) is \textbf{not} rejected when it is \textbf{false}
+ \end{description}
+
+% \subsection*{Using c.i. to find \(p\)}
+% need more here
+
\end{document}