\usepackage{xcolor} % used only to show the phantomed stuff
\definecolor{cas}{HTML}{e6f0fe}
\usepackage{mathtools}
+\pgfplotsset{compat=1.16}
\pagestyle{fancy}
\fancyhead[LO,LE]{Unit 4 Specialist --- Statistics}
\begin{align*}
\Pr(Y \le y) &= \Pr(aX+b \le y) \\
&= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
- &= \int^{\dfrac{y-b}{a}}_{-\infty} f(x) \> dx
+ &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx
\end{align*}
\begin{align*}
\section{Sample mean}
+ Approximation of the \textbf{population mean} determined experimentally.
+
\[ \overline{x} = \dfrac{\Sigma x}{n} \]
- where \(n\) is the size of the sample (number of sample points)
+ where \(n\) is the size of the sample (number of sample points) and \(x\) is the value of a sample point
- \subsubsection*{\colorbox{cas}{On CAS:}}
+ \begin{tcolorbox}[colframe=cas!75!black, title=On CAS]
\begin{enumerate}
\item Spreadsheet
\item Input range as A1:An where \(n\) is the number of samples
\item Graph \(\rightarrow\) Histogram
\end{enumerate}
+ \end{tcolorbox}
\subsubsection*{Sample size of \(n\)}
\[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
- Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\)
-
+ Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).
+ \begin{tcolorbox}[colframe=cas!75!black, title=On CAS]
+
+ \begin{itemize}
+ \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left
+ \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
+ \end{itemize}
+ \end{tcolorbox}
+
+ \section{Normal distributions}
+
+ mean = mode = median
+
+ \[ Z = \frac{X - \mu}{\sigma} \]
+
+ Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\)
+\pgfmathdeclarefunction{gauss}{2}{%
+ \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
+}
+
+{\begin{center} \begin{tikzpicture}
+ \pgfplotsset{set layers}
+\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={\((\mu-2\sigma)\), \((\mu-\sigma)\), \(\mu\), \((\mu+\sigma)\), \((\mu+2\sigma)\)},
+ xlabel={\(x\)},
+ every axis x label/.style={at={(current axis.right of origin)},anchor=north west},
+ every axis y label/.style={at={(axis description cs:-0.02,0.2)}, anchor=south west, rotate=90},
+ ylabel={\(\Pr(X=x)\)}]
+ \addplot {gauss(0,0.75)};
+\end{axis}
+\begin{axis}[every axis plot post/.append style={
+ mark=none,domain=-3:3,samples=50,smooth},
+ axis x line=bottom,
+ enlargelimits=upper,
+ x=\textwidth/10,
+ xtick={-2,-1,0,1,2},
+ axis x line shift=30pt,
+ hide y axis,
+ x tick label style = {font=\footnotesize},
+ xlabel={\(Z\)},
+ every axis x label/.style={at={(axis description cs:1,-0.25)},anchor=south west}]
+ \addplot {gauss(0,0.75)};
+\end{axis}
+\end{tikzpicture}\end{center}}
+
+ \section{Central limit theorem}
+
+ If \(X\) is randomly distributed with mean \(\mu\) and sd \(\sigma\), then with an adequate sample size \(n\) the distribution of the sample mean \(\overline{X}\) is approximately normal with mean \(E(\overline{X})\) and \(\operatorname{sd}(\overline{X}) = \frac{\sigma}{\sqrt{n}}\).
\end{document}