\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}
\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:}}
\[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
- 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 \(n\)).
+ \colorbox{cas}{On CAS:} 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 \\
+ To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
+
+ \section{Normal distributions}
+
+ mean = mode = median
+
+ \[ Z = \frac{X - \mu}{\sigma} \]
+
+ Normal distributions must have are (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{tikzpicture}
+\begin{axis}[every axis plot post/.append style={
+ mark=none,domain=-3:3,samples=50,smooth}, % All plots: from -2:2, 50 samples, smooth, no marks
+ axis x line*=bottom, % no box around the plot, only x and y axis
+ axis y line*=left, % the * suppresses the arrow tips
+ enlargelimits=upper,
+ ytick={0.5},
+ yticklabels={\(\frac{1}{\sigma \sqrt{2\pi}}\)},
+ xtick={-2,-1,0,1,2},
+ 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},
+ ylabel={\(\Pr(X=x)\)}]
+ \addplot {gauss(0,0.75)};
+\end{axis}
+\end{tikzpicture}
+
+ \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}