[spec] reorganise notes for sac

diff --git a/spec/dynamics.pdf b/spec/dynamics.pdf
index ca06d3747805d82297d78caf4cf3f3f531790f1d..dc59835e128918c05f5f7faeb3a93f4cea6c7a8f 100644 (file)
Binary files a/spec/dynamics.pdf and b/spec/dynamics.pdf differ

diff --git a/spec/dynamics.tex b/spec/dynamics.tex
index 5891f4e7195bc967116ba2e5f1aaf8e22b49f49b..b6a1af8cd718e464c8be211f60add75fc9525b33 100644 (file)
--- a/spec/dynamics.tex
+++ b/spec/dynamics.tex
@@ -1,105 +1,58 @@
-\documentclass[a4paper, tikz, pstricks]{article}
-\usepackage[a4paper,margin=2cm]{geometry}
-\usepackage{array}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{tcolorbox}
-\usepackage{fancyhdr}
-\usepackage{pgfplots}
-\usepackage{tikz}
-\usetikzlibrary{arrows,
-    calc,
-    decorations,
-    scopes,
-    angles
-}
-\usetikzlibrary{calc}
-\usetikzlibrary{angles}
-\usetikzlibrary{datavisualization.formats.functions}
-\usetikzlibrary{decorations.markings}
-\usepgflibrary{arrows.meta}
-\usetikzlibrary{decorations.markings}
-\usepgflibrary{arrows.meta}
-\usepackage{pst-plot}
-\psset{dimen=monkey,fillstyle=solid,opacity=.5}
-\def\object{%
-    \psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1)
-    \psaxes[linecolor=gray,labels=none,ticks=none]{->}(0,0)(-3,-3)(3,2)[$x$,0][$y$,90]
-    \rput{*0}{%
-        \psline{->}(0,-2)%
-        \uput[-90]{*0}(0,-2){$\vec{w}$}}
-}
-
-\usepackage{tabularx}
-\usetikzlibrary{angles}
-\usepackage{keystroke}
-\usepackage{listings}
-\usepackage{xcolor} % used only to show the phantomed stuff
-\definecolor{cas}{HTML}{e6f0fe}
-
-\pagestyle{fancy}
-\fancyhead[LO,LE]{Year 12 Specialist - Dynamics}
-\fancyhead[CO,CE]{Andrew Lorimer}
-
-\setlength\parindent{0pt}
-
+\documentclass[spec-collated.tex]{subfiles}

 \begin{document}
 
 \begin{document}
 

-\title{Dynamics}
-\author{}
-\date{}
-\maketitle
+\section{Dynamics}

 
 

-\section{Resolution of forces}
+\subsection*{Resolution of forces}

 
 \textbf{Resultant force} is sum of force vectors
 
 
 \textbf{Resultant force} is sum of force vectors
 

-\subsection*{In angle-magnitude form}
+\subsubsection*{In angle-magnitude form}

 
 \makebox[3cm]{Cosine rule:} \(c^2=a^2+b^2-2ab\cos\theta\)
 \makebox[3cm]{Sine rule:} \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)
 
 
 \makebox[3cm]{Cosine rule:} \(c^2=a^2+b^2-2ab\cos\theta\)
 \makebox[3cm]{Sine rule:} \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)
 

-\subsection*{In \(\boldsymbol{i}\)---\(\boldsymbol{j}\) form}
+\subsubsection*{In \(\boldsymbol{i}\)---\(\boldsymbol{j}\) form}

 
 Vector of \(a\) N at \(\theta\) to \(x\) axis is equal to \(a \cos \theta \boldsymbol{i} + a \sin \theta \boldsymbol{j}\). Convert all force vectors then add.
 
 To find angle of an \(a\boldsymbol{i} + b\boldsymbol{j}\) vector, use \(\theta = \tan^{-1} \frac{b}{a}\)
 
 
 Vector of \(a\) N at \(\theta\) to \(x\) axis is equal to \(a \cos \theta \boldsymbol{i} + a \sin \theta \boldsymbol{j}\). Convert all force vectors then add.
 
 To find angle of an \(a\boldsymbol{i} + b\boldsymbol{j}\) vector, use \(\theta = \tan^{-1} \frac{b}{a}\)
 

-\subsection*{Resolving in a given direction}
+\subsubsection*{Resolving in a given direction}

 
 The resolved part of a force \(P\) at angle \(\theta\) is has magnitude \(P \cos \theta\)
 
 To convert force \(||\vec{OA}\) to angle-magnitude form, find component \(\perp\vec{OA}\) then \(|\boldsymbol{r}|=\sqrt{\left(||\vec{OA}\right)^2 + \left(\perp\vec{OA}\right)^2},\quad \theta = \tan^{-1}\dfrac{\perp\vec{OA}}{||\vec{OA}}\)
 
 
 The resolved part of a force \(P\) at angle \(\theta\) is has magnitude \(P \cos \theta\)
 
 To convert force \(||\vec{OA}\) to angle-magnitude form, find component \(\perp\vec{OA}\) then \(|\boldsymbol{r}|=\sqrt{\left(||\vec{OA}\right)^2 + \left(\perp\vec{OA}\right)^2},\quad \theta = \tan^{-1}\dfrac{\perp\vec{OA}}{||\vec{OA}}\)
 

-\section{Newton's laws}
+\subsection*{Newton's laws}

 
 \begin{tcolorbox}
 
 \begin{tcolorbox}

-  \begin{enumerate}
-    \item Velocity is constant without a net external velocity
+  \begin{enumerate}[leftmargin=1mm]
+    \item Velocity is constant without a net external force

     \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\)
     \item Equal and opposite forces
   \end{enumerate}
 \end{tcolorbox}
 
     \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\)
     \item Equal and opposite forces
   \end{enumerate}
 \end{tcolorbox}
 

-\subsection*{Weight}
+\subsubsection*{Weight}

 A mass of \(m\) kg has force of \(mg\) acting on it
 
 A mass of \(m\) kg has force of \(mg\) acting on it
 

-\subsection*{Momentum \(\rho\)}
+\subsubsection*{Momentum \(\rho\)}

 \[ \rho = mv \tag{units kg m/s or Ns} \]
 
 \[ \rho = mv \tag{units kg m/s or Ns} \]
 

-\subsection*{Reaction force \(R\)}
+\subsubsection*{Reaction force \(R\)}

 
 \begin{itemize}
   \item With no vertical velocity, \(R=mg\)
 
 \begin{itemize}
   \item With no vertical velocity, \(R=mg\)

-  \item With upwards acceleration, \(R-mg=ma\)
+  \item With vertical acceleration, \(|R|=m|a|-mg\)

   \item With force \(F\) at angle \(\theta\), then \(R=mg-F\sin\theta\)
 \end{itemize}
 
   \item With force \(F\) at angle \(\theta\), then \(R=mg-F\sin\theta\)
 \end{itemize}
 

-\subsection*{Friction}
+\subsubsection*{Friction}

 
 \[ F_R = \mu R \tag{friction coefficient} \]
 
 
 \[ F_R = \mu R \tag{friction coefficient} \]
 

-\section{Inclined planes}
+\subsection*{Inclined planes}

 
 \[ \boldsymbol{F} = |\boldsymbol{F}| \cos \theta \boldsymbol{i} + |\boldsymbol{F}| \sin \theta \boldsymbol{j} \]
 \begin{itemize}
 
 \[ \boldsymbol{F} = |\boldsymbol{F}| \cos \theta \boldsymbol{i} + |\boldsymbol{F}| \sin \theta \boldsymbol{j} \]
 \begin{itemize}


@@ -122,9 +75,7 @@ A mass of \(m\) kg has force of \(mg\) acting on it
   pulley/.style={thick}
 }
 
   pulley/.style={thick}
 }
 

-\begin{figure}[!htb]
-  \centering
-  \begin{tikzpicture}
+  \begin{center}\begin{tikzpicture}

 
     \pgfmathsetmacro{\Fnorme}{2}
     \pgfmathsetmacro{\Fangle}{30}
 
     \pgfmathsetmacro{\Fnorme}{2}
     \pgfmathsetmacro{\Fangle}{30}


@@ -153,17 +104,9 @@ A mass of \(m\) kg has force of \(mg\) acting on it
     \end{scope}
     \draw[force,->] (M.center) -- ++(0,-1) node[below] {$mg$};
     \draw (M.center)+(-90:\arcr) arc [start angle=-90,end angle=\iangle-90,radius=\arcr] node [below, pos=.5] {\footnotesize\(\theta\)};
     \end{scope}
     \draw[force,->] (M.center) -- ++(0,-1) node[below] {$mg$};
     \draw (M.center)+(-90:\arcr) arc [start angle=-90,end angle=\iangle-90,radius=\arcr] node [below, pos=.5] {\footnotesize\(\theta\)};

-  \end{tikzpicture}
-\end{figure}
+  \end{tikzpicture}\end{center}

 
 

-\section{Connected particles}
-
-\begin{itemize}
-  \item \textbf{Suspended pulley:} tension in both sections of rope are equal
-  \item \textbf{Linear connection:} find acceleration of system first
-  \item \textbf{Pulley on right angle:} \(a = \frac{m_2g}{m_1+m_2}\) where \(m_2\) is suspended (frictionless on both surfaces)
-  \item \textbf{Pulley on edge of incline:} find downwards force \(W_2\) and components of mass on plane
-\end{itemize}
+\subsection*{Connected particles}

 
 \def\boxwidth{0.5}
 \tikzset{
 
 \def\boxwidth{0.5}
 \tikzset{


@@ -172,8 +115,7 @@ A mass of \(m\) kg has force of \(mg\) acting on it
 }
 
 
 }
 
 

-\begin{figure}[!htb]
-  \centering
+\begin{center}

   \begin{tikzpicture}
 
     \matrix[column sep=1cm] {
   \begin{tikzpicture}
 
     \matrix[column sep=1cm] {


@@ -231,22 +173,68 @@ A mass of \(m\) kg has force of \(mg\) acting on it
       \\
     };
   \end{tikzpicture}
       \\
     };
   \end{tikzpicture}

-\end{figure}
+  \end{center}
+
+\begin{itemize}
+  \item \textbf{Suspended pulley:} tension in both sections of rope are equal \\
+    \(|a| = g \frac{m_1 - m_2}{m_1 + m_2}\) where \(m_1\) accelerates down \\
+    With tension: \\
+    \[ \begin{cases}m_1 g - T = m_1 a\\ T - m_2 g = m_2 a\end{cases} \\ \implies m_1 g - m_2 g = m_1 a + m_2 a \]
+  \item \textbf{String pulling mass on inclined pane:} Resolve parallel to plane \\
+    \[ T-mg \sin \theta = ma \]
+  \item \textbf{Linear connection:} find acceleration of system first
+  \item \textbf{Pulley on right angle:} \(a = \frac{m_2g}{m_1+m_2}\) where \(m_2\) is suspended (frictionless on both surfaces)
+  \item \textbf{Pulley on edge of incline:} find downwards force \(W_2\) and components of mass on plane
+\end{itemize}
+
+\hspace{2em}\parbox{8em}{In this example, note \(T_1 \ne T_2\):}
+  \begin{tikzpicture}
+
+      \begin{scope}

 
 

-\section{Equilibrium}
+        \coordinate (O) at (0,0);
+        \coordinate (A) at ($({3*cos(\iangle)},{3*sin(\iangle)})$);
+        \coordinate (B) at ($({3*cos(\iangle)},0)$);
+        \coordinate (C) at ($({(1-0.25*\boxwidth)*cos(\iangle)},{(1-0.25*\boxwidth)*sin(\iangle)})$); % centre of box
+        \coordinate (D) at ($(C)+(\iangle:\boxwidth)$);
+        \coordinate (E) at ($(D)+(90+\iangle:0.5*\boxwidth)$);
+        \coordinate (F) at ($(B)+(0,{1.5*sin(\iangle)})$);
+        \coordinate (G) at ($(A)+(\iangle:-2*\boxwidth)$);
+        \coordinate (H) at ($(G)+(90+\iangle:0.5*\boxwidth)$);
+        \coordinate (I) at ($(H)+(\iangle:-0.5*\boxwidth)$);
+        \coordinate (J) at ($(H)+(\iangle:\boxwidth)$);
+        \coordinate (X) at ($(A)+(\iangle:0.5*\boxwidth)$); % centre of pulley
+        \coordinate (Y) at ($(X)+(90+\iangle:0.5*\boxwidth)$); % chord of pulley
+
+        \draw[plane] (O) -- (A) -- (B) -- (O);
+        \draw (O)+(\arcr,0) arc [start angle=0,end angle=\iangle,radius=\arcr] node [right, pos=.75] {\footnotesize\(\theta\)};
+
+        \draw [rotate=\iangle, m] (C) rectangle ++(\boxwidth,\boxwidth) node (z) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
+        \draw [rotate=\iangle, m] (G) rectangle ++(\boxwidth,\boxwidth) node (l) [rotate=\iangle, midway, font=\footnotesize] {\(m_2\)};
+        \draw [pulley] (A) -- (X) ++(0.5*\boxwidth, 0) arc[rotate=\iangle, start angle=0, delta angle=360, x radius=0.25, y radius=0.25] node(r) [midway, rotate=\iangle] {};
+        \draw [string] (E) -- (H) node [midway, above, font=\footnotesize, rotate=\iangle] {\(T_2\)};
+        \draw [string] (J) -- (Y) node [midway, above, font=\footnotesize, rotate=\iangle] {\(T_1\)} arc (90+\iangle:0:0.25) -- ++($(0,{-1.5*sin(\iangle)})$) node [midway, above right, font=\footnotesize] {\(T_1\)} node[m] {\(m_3\)};
+
+      \end{scope}
+
+  \end{tikzpicture}
+\subsection*{Equilibrium}

 
 \[ \dfrac{A}{\sin a} = \dfrac{B}{\sin b} = \dfrac{C}{\sin c} \tag{Lami's theorem}\]
 
 \[ \dfrac{A}{\sin a} = \dfrac{B}{\sin b} = \dfrac{C}{\sin c} \tag{Lami's theorem}\]

+\[ c^2 = a^2 + b^2 - 2ab \cos \theta \tag{cosine rule} \]

 
 Three methods:
 \begin{enumerate}
   \item Lami's theorem (sine rule)
 
 Three methods:
 \begin{enumerate}
   \item Lami's theorem (sine rule)

-  \item Triangle of forces or CAS (use to verify)
+  \item Triangle of forces (cosine rule)

   \item Resolution of forces (\(\Sigma F = 0\) - simultaneous)
 \end{enumerate}
 
   \item Resolution of forces (\(\Sigma F = 0\) - simultaneous)
 \end{enumerate}
 

-\colorbox{cas}{On CAS:} use Geometry, lock known constants.
+  \begin{cas}
+    \textbf{To verify:} Geometry tab, then select points with normal cursor. Click right arrow at end of toolbar and input point, then lock known constants.
+  \end{cas}

 
 

-\section{Variable forces (DEs)}
+\subsection*{Variable forces (DEs)}

 
 \[ a = \dfrac{d^2x}{dt^2} = \dfrac{dv}{dt} = v\dfrac{dv}{dx} = \dfrac{d}{dx} \left( \frac{1}{2} v^2 \right) \]
 
 
 \[ a = \dfrac{d^2x}{dt^2} = \dfrac{dv}{dt} = v\dfrac{dv}{dx} = \dfrac{d}{dx} \left( \frac{1}{2} v^2 \right) \]
 

diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf
index 2af5693473797d195b23dfaba30384137b9a4b4f..7aa2159ebf421d89567fbb2924c33b3875b71986 100644 (file)
Binary files a/spec/spec-collated.pdf and b/spec/spec-collated.pdf differ

diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex
index 63d724dd94af056e9530db74371218f8dc936883..0525085edb4a24ea2b82239d61b478374fc9ce3b 100644 (file)
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -1,12 +1,15 @@
 \documentclass[a4paper]{article}
 \usepackage[a4paper,margin=2cm]{geometry}
 \usepackage{multicol}
 \documentclass[a4paper]{article}
 \usepackage[a4paper,margin=2cm]{geometry}
 \usepackage{multicol}

+\usepackage{dblfloatfix}

 \usepackage{multirow}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{harpoon}
 \usepackage{tabularx}
 \usepackage{makecell}
 \usepackage{multirow}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{harpoon}
 \usepackage{tabularx}
 \usepackage{makecell}

+\usepackage{enumitem}
+\usepackage[obeyspaces]{url}

 \usepackage[dvipsnames, table]{xcolor}
 \usepackage{blindtext}
 \usepackage{graphicx}
 \usepackage[dvipsnames, table]{xcolor}
 \usepackage{blindtext}
 \usepackage{graphicx}


@@ -15,6 +18,27 @@
 \usepackage{tkz-fct}
 \usepackage{tikz-3dplot}
 \usepackage{pgfplots}
 \usepackage{tkz-fct}
 \usepackage{tikz-3dplot}
 \usepackage{pgfplots}

+\usetikzlibrary{arrows,
+    decorations,
+    decorations.markings,
+    decorations.text,
+    scopes
+}
+\usetikzlibrary{datavisualization.formats.functions}
+\usetikzlibrary{decorations.markings}
+\usepgflibrary{arrows.meta}
+\usetikzlibrary{decorations.markings}
+\usepgflibrary{arrows.meta}
+\usepackage{pst-plot}
+\psset{dimen=monkey,fillstyle=solid,opacity=.5}
+\def\object{%
+    \psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1)
+    \psaxes[linecolor=gray,labels=none,ticks=none]{->}(0,0)(-3,-3)(3,2)[$x$,0][$y$,90]
+    \rput{*0}{%
+        \psline{->}(0,-2)%
+        \uput[-90]{*0}(0,-2){$\vec{w}$}}
+}
+

 \usetikzlibrary{calc}
 \usetikzlibrary{angles}
 \usetikzlibrary{datavisualization.formats.functions}
 \usetikzlibrary{calc}
 \usetikzlibrary{angles}
 \usetikzlibrary{datavisualization.formats.functions}


@@ -24,7 +48,6 @@
 \pagestyle{fancy}
 \fancyhead[LO,LE]{Year 12 Specialist}
 \fancyhead[CO,CE]{Andrew Lorimer}
 \pagestyle{fancy}
 \fancyhead[LO,LE]{Year 12 Specialist}
 \fancyhead[CO,CE]{Andrew Lorimer}

-

 \usepackage{mathtools}
 \usepackage{xcolor} % used only to show the phantomed stuff
 \renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
 \usepackage{mathtools}
 \usepackage{xcolor} % used only to show the phantomed stuff
 \renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!


@@ -34,21 +57,31 @@
 \newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}%
 \newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}%
 \definecolor{cas}{HTML}{e6f0fe}
 \newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}%
 \newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}%
 \definecolor{cas}{HTML}{e6f0fe}

+\definecolor{important}{HTML}{fc9871}
+\definecolor{dark-gray}{gray}{0.2}

 \linespread{1.5}
 \newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
 \newcommand{\tg}{\mathop{\mathrm{tg}}}
 \newcommand{\cotg}{\mathop{\mathrm{cotg}}}
 \newcommand{\arctg}{\mathop{\mathrm{arctg}}}
 \newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
 \linespread{1.5}
 \newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
 \newcommand{\tg}{\mathop{\mathrm{tg}}}
 \newcommand{\cotg}{\mathop{\mathrm{cotg}}}
 \newcommand{\arctg}{\mathop{\mathrm{arctg}}}
 \newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}

-
-
-                  \pgfplotsset{every axis/.append style={
-                    axis x line=middle,    % put the x axis in the middle
-                    axis y line=middle,    % put the y axis in the middle
-                    axis line style={->}, % arrows on the axis
-                    xlabel={$x$},          % default put x on x-axis
-                    ylabel={$y$},          % default put y on y-axis
-                  }}
+\pgfplotsset{every axis/.append style={
+  axis x line=middle,    % put the x axis in the middle
+  axis y line=middle,    % put the y axis in the middle
+  axis line style={->}, % arrows on the axis
+  xlabel={$x$},          % default put x on x-axis
+  ylabel={$y$},          % default put y on y-axis
+}}
+\usepackage{tcolorbox}
+\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries}
+\newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm}
+\usepackage{keystroke}
+\usepackage{listings}
+\usepackage{mathtools}
+\pgfplotsset{compat=1.16}
+\usepackage{subfiles}
+\usepackage{import}
+\setlength{\parindent}{0pt}

 \begin{document}
 
 \begin{multicols}{2}
 \begin{document}
 
 \begin{multicols}{2}


@@ -863,10 +896,6 @@
                   \(f^{\prime\prime} = 0\))
 
 
                   \(f^{\prime\prime} = 0\))
 
 

-                  \pgfplotsset{every axis/.append style={
-                    axis x line=none,    % put the x axis in the middle
-                    axis y line=none,    % put the y axis in the middle
-                  }}

                   \begin{table*}[ht]
                     \centering
                     \begin{tabularx}{\textwidth}{rXXX}
                   \begin{table*}[ht]
                     \centering
                     \begin{tabularx}{\textwidth}{rXXX}


@@ -875,18 +904,18 @@
                       & \centering\(\dfrac{d^2 y}{dx^2} > 0\)  & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
                       \hline
                       \(\dfrac{dy}{dx}>0\) &
                       & \centering\(\dfrac{d^2 y}{dx^2} > 0\)  & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
                       \hline
                       \(\dfrac{dy}{dx}>0\) &

-                      \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-3,  xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x))};  \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}&
-                        \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=0.1, xmax=4,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))};  \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}&
-                          \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1.5,  xmax=1.5,   scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\
+                      \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-3,  xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x))};  \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}&
+                        \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0.1, xmax=4,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))};  \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}&
+                          \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1.5,  xmax=1.5,   scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\

                             \hline
                             \(\dfrac{dy}{dx}<0\) &
                             \hline
                             \(\dfrac{dy}{dx}<0\) &

-                            \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {(1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}&
-                              \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=0,  xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)};  \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}&
-                                \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=1.5,  xmax=4.5,   scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\
+                            \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {(1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}&
+                              \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0,  xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)};  \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}&
+                                \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=1.5,  xmax=4.5,   scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\

                                   \hline
                                   \(\dfrac{dy}{dx}=0\)&
                                   \hline
                                   \(\dfrac{dy}{dx}=0\)&

-                                  \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}&                       \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x))}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}&
-                                    \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture}  \\Stationary inflection point}\\
+                                  \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}&                       \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x))}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}&
+                                    \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture}  \\Stationary inflection point}\\

                                       \hline
                     \end{tabularx}
                   \end{table*}
                                       \hline
                     \end{tabularx}
                   \end{table*}


@@ -1217,5 +1246,7 @@
         Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
         \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
 
         Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
         \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
 

+      \subfile{dynamics}
+      \subfile{statistics}

   \end{multicols}
 \end{document}
   \end{multicols}
 \end{document}

diff --git a/spec/statistics.pdf b/spec/statistics.pdf
index 60f093cbb5ef448216735265cda25af976f03e68..9a63860bef32597f9e026bdcc95c2915da78773d 100644 (file)
Binary files a/spec/statistics.pdf and b/spec/statistics.pdf differ

diff --git a/spec/statistics.tex b/spec/statistics.tex
index 4c89dd596bf579f9ddbea537a1039c46443e3641..da1ec9b7711634155706c8d01b221754fa53c06b 100644 (file)
--- a/spec/statistics.tex
+++ b/spec/statistics.tex
@@ -1,33 +1,7 @@
-\documentclass[a4paper]{article}
-\usepackage[a4paper, margin=2cm]{geometry}
-\usepackage{array}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{tcolorbox}
-\usepackage{fancyhdr}
-\usepackage{pgfplots}
-\usepackage{tabularx}
-\usepackage{keystroke}
-\usepackage{listings}
-\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}
-\fancyhead[CO,CE]{Andrew Lorimer}
-
-\setlength\parindent{0pt}
-
+\documentclass[spec-collated.tex]{subfiles}

 \begin{document}
 
 \begin{document}
 

-  \title{Statistics}
-  \author{}
-  \date{}
-  \maketitle
-
-  \section{Linear combinations of random variables}
+  \section{Statistics}

 
   \subsection*{Continuous random variables}
 
 
   \subsection*{Continuous random variables}
 


@@ -38,9 +12,23 @@
     \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
   \end{enumerate}
 
     \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
   \end{enumerate}
 

+  \begin{align*}
+    E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\
+    \operatorname{Var}(X) &= E\left[(X-\mu)^2\right]
+  \end{align*}
+

   \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
   \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]

+  

 
 

-  \subsubsection*{Linear functions \(X \rightarrow aX+b\)}
+  \subsection*{Two random variables \(X, Y\)}
+
+  If \(X\) and \(Y\) are independent:
+  \begin{align*}
+    \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
+    \operatorname{Var}(aX \pm bY \pm c) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y)
+  \end{align*}
+
+  \subsection*{Linear functions \(X \rightarrow aX+b\)}

 
   \begin{align*}
     \Pr(Y \le y) &= \Pr(aX+b \le y) \\
 
   \begin{align*}
     \Pr(Y \le y) &= \Pr(aX+b \le y) \\


@@ -53,31 +41,40 @@
     \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
   \end{align*}
 
     \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
   \end{align*}
 

-  \subsection*{Linear combination of two random variables}
+  \subsection*{Expectation theorems}
+
+  For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).

 
   \begin{align*}
 
   \begin{align*}

-    \textbf{Mean:} && \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
-    \textbf{Variance:} && \operatorname{Var}(aX+bY) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) \tag{if \(X\) and \(Y\) are independent}\\
+    E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\
+    E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\
+    &\ne [E(X)]^n \\
+    E(aX \pm b) &= aE(X) \pm b \tag{linear} \\
+    E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\
+    E(X+Y) &= E(X) + E(Y) \tag{two variables}

   \end{align*}
 
   \end{align*}
 

-  \section{Sample mean}
+  \subsection*{Sample mean}

 
   Approximation of the \textbf{population mean} determined experimentally.
 
   \[ \overline{x} = \dfrac{\Sigma x}{n} \]
 
 
   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) and \(x\) is the value of a sample point
-
-  \begin{tcolorbox}[colframe=cas!75!black, title=On CAS]
+  where
+  \begin{description}[nosep, labelindent=0.5cm]
+    \item \(n\) is the size of the sample (number of sample points)
+    \item \(x\) is the value of a sample point
+  \end{description}

 
 

-  \begin{enumerate}
+\begin{cas}
+  \begin{enumerate}[leftmargin=3mm]

     \item Spreadsheet
     \item Spreadsheet

-    \item In cell A1: \verb;mean(randNorm(sd, mean, sample size));
+    \item In cell A1:\\ \path{mean(randNorm(sd, mean, sample size))}

     \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
     \item Input range as A1:An where \(n\) is the number of samples
     \item Graph \(\rightarrow\) Histogram
   \end{enumerate}
     \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
     \item Input range as A1:An where \(n\) is the number of samples
     \item Graph \(\rightarrow\) Histogram
   \end{enumerate}

-  \end{tcolorbox}
+  \end{cas}

 
   \subsubsection*{Sample size of \(n\)}
 
 
   \subsubsection*{Sample size of \(n\)}
 


@@ -87,27 +84,60 @@
 
   For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)
 
 
   For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)
 

-  \begin{tcolorbox}[colframe=cas!75!black, title=On CAS]
+  \begin{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}
   
     \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}
+
+  \end{cas}

   
   

-  \section{Normal distributions}
+  \subsection*{Normal distributions}

 
 

-  mean = mode = median

 
   \[ Z = \frac{X - \mu}{\sigma} \]
 
 
   \[ Z = \frac{X - \mu}{\sigma} \]
 

-  Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\)
+  Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\
+  \(\text{mean} = \text{mode} = \text{median}\)
+
+  \begin{warning}
+    Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
+  \end{warning}
+

 \pgfmathdeclarefunction{gauss}{2}{%
 \pgfmathdeclarefunction{gauss}{2}{%

-  \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}
+  \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%

 }
 }

-
-\begin{tikzpicture}
-  \pgfplotsset{set layers}
+                  \pgfplotsset{every axis/.append style={
+                    axis x line=middle,    % put the x axis in the middle
+                    axis y line=middle,    % put the y axis in the middle
+                  }} \pgfkeys{/pgf/decoration/.cd,
+      distance/.initial=10pt
+}  \pgfdeclaredecoration{add dim}{final}{
+\state{final}{% 
+\pgfmathsetmacro{\dist}{5pt*\pgfkeysvalueof{/pgf/decoration/distance}/abs(\pgfkeysvalueof{/pgf/decoration/distance})}    
+          \pgfpathmoveto{\pgfpoint{0pt}{0pt}}             
+          \pgfpathlineto{\pgfpoint{0pt}{2*\dist}}   
+          \pgfpathmoveto{\pgfpoint{\pgfdecoratedpathlength}{0pt}} 
+          \pgfpathlineto{\pgfpoint{(\pgfdecoratedpathlength}{2*\dist}}     
+          \pgfsetarrowsstart{latex}
+          \pgfsetarrowsend{latex}
+          \pgfpathmoveto{\pgfpoint{0pt}{\dist}}
+          \pgfpathlineto{\pgfpoint{\pgfdecoratedpathlength}{\dist}} 
+          \pgfusepath{stroke} 
+          \pgfpathmoveto{\pgfpoint{0pt}{0pt}}
+          \pgfpathlineto{\pgfpoint{\pgfdecoratedpathlength}{0pt}}
+}}
+\tikzset{dim/.style args={#1,#2}{decoration={add dim,distance=#2},
+                decorate,
+                postaction={decorate,decoration={text along path,
+                                                 raise=#2,
+                                                 text align={align=center},
+                                                 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, 
 \begin{axis}[every axis plot post/.append style={
   mark=none,domain=-3:3,samples=50,smooth}, 
   axis x line=bottom, 


@@ -124,6 +154,24 @@
   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)};
   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)};

+\fill[red!30] (-3,0)  -- plot[id=f3,domain=-3:3,samples=50]
+        function {1/(0.75*sqrt(2*pi))*exp(-((x)^2)/(2*0.75^2))} -- (3,0) -- cycle;
+  \fill[darkgray!30] (3,0)  -- plot[id=f3,domain=-3:3,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (3,0) -- cycle;
+  \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;
+  \fill[white!30] (-1,0)  -- plot[id=f3,domain=-1:1,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (1,0) -- cycle;
+  \begin{scope}[<->]
+    \draw (-1,0.35) -- (1,0.35) node [midway, fill=white] {68.3\%};
+    \draw (-2,0.25) -- (2,0.25) node [midway, fill=white] {95.5\%};
+    \draw (-3,0.15) -- (3,0.15) node [midway, fill=white] {99.7\%};
+  \end{scope}
+  \begin{scope}[-, dashed, gray]
+    \draw (-1,0) -- (-1, 0.35);
+    \draw (1,0) -- (1, 0.35);
+    \draw (-2,0) -- (-2, 0.25);
+    \draw (2,0) -- (2, 0.25);
+    \draw (-3,0) -- (-3, 0.15);
+    \draw (3,0) -- (3, 0.15);
+  \end{scope}

 \end{axis}
 \begin{axis}[every axis plot post/.append style={
   mark=none,domain=-3:3,samples=50,smooth}, 
 \end{axis}
 \begin{axis}[every axis plot post/.append style={
   mark=none,domain=-3:3,samples=50,smooth}, 


@@ -138,55 +186,55 @@
   every axis x label/.style={at={(axis description cs:1,-0.25)},anchor=south west}]
   \addplot {gauss(0,0.75)};
 \end{axis}
   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{tikzpicture}\end{center}}
+                  \end{figure*}

 
 

-  \section{Central limit theorem}
+  \subsection*{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 \(\operatorname{sd}(\overline{X}) = \frac{\sigma}{\sqrt{n}}\).
 
 
   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 \(\operatorname{sd}(\overline{X}) = \frac{\sigma}{\sqrt{n}}\).
 

-  \section{Confidence intervals}
+  \subsection*{Confidence intervals}

 
   \begin{itemize}
     \item \textbf{Point estimate:} single-valued estimate of the population mean from the value of the sample mean \(\overline{x}\)
     \item \textbf{Interval estimate:} confidence interval for population mean \(\mu\)
 
   \begin{itemize}
     \item \textbf{Point estimate:} single-valued estimate of the population mean from the value of the sample mean \(\overline{x}\)
     \item \textbf{Interval estimate:} confidence interval for population mean \(\mu\)

+    \item \(C\)\% confidence interval \(\implies\) \(C\)\% of samples will contain population mean \(\mu\)

   \end{itemize}
 
   \end{itemize}
 

-  \subsection{95\% confidence interval}
-
-  The 95\% confidence interval for a population mean \(\mu\) is given by
-
-  \[ \overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \]
-
-  where: \\
-  \(\overline{x}\) is the sample mean \\
-  \(\sigma\) is the population sd \\
-  \(n\) is the sample size from which \(\overline{x}\) was calculated
+  \subsubsection*{95\% confidence interval}

 
 

-  Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
+  For 95\% c.i. of population mean \(\mu\):

 
 

-  \colorbox{cas}{\textbf{On CAS}}
+  \[ x \in \left(\overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \right)\]

 
 

-  Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
-  Set Type = One-Sample Z Int, Variable
+  where:
+  \begin{description}[nosep, labelindent=0.5cm]
+    \item \(\overline{x}\) is the sample mean
+    \item \(\sigma\) is the population sd
+    \item \(n\) is the sample size from which \(\overline{x}\) was calculated
+  \end{description}

 
 

-  \subsection*{Interpretation of confidence intervals}
-
-  95\% confidence interval \(\implies\) 95\% of samples will contain population mean \(\mu\).
+  \begin{cas}
+    Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
+    Set \textit{Type = One-Sample Z Int} \\ \-\hspace{1em} and select \textit{Variable}
+  \end{cas}

 
   \subsection*{Margin of error}
 
 
   \subsection*{Margin of error}
 

-  For 95\% confidence interval for \(\mu\), margin of error \(M\) is:
-
+  For 95\% confidence interval of \(\mu\):

   \begin{align*}
     M &= 1.96 \times \dfrac{\sigma}{\sqrt{n}} \\
     \implies n &= \left( \dfrac{1.96 \sigma}{M} \right)^2
   \end{align*}
 
   \begin{align*}
     M &= 1.96 \times \dfrac{\sigma}{\sqrt{n}} \\
     \implies n &= \left( \dfrac{1.96 \sigma}{M} \right)^2
   \end{align*}
 

+  Always round \(n\) up to a whole number of samples.
+

   \subsection*{General case}
 
   \subsection*{General case}
 

-  A confidence interval of \(C\)\% for a mean \(\mu\)  s given by
+  For \(C\)\% c.i. of population mean \(\mu\):

 
 

-  \[ x \in \left( \overline{x} \pm k \dfrac{\sigma}{\sqrt{n}} \right) \quad \text{ where } k \text{ is such that } \Pr(-k < Z < k) = \frac{C}{100} \]
+  \[ x \in \left( \overline{x} \pm k \dfrac{\sigma}{\sqrt{n}} \right) \]
+  \hfill where \(k\) is such that \(\Pr(-k < Z < k) = \frac{C}{100}\)

 
   \subsection*{Confidence interval for multiple trials}
 
 
   \subsection*{Confidence interval for multiple trials}
 


@@ -194,7 +242,9 @@
 
   \section{Hypothesis testing}
 
 
   \section{Hypothesis testing}
 

-  Note hypotheses are always expressed in terms of population parameters
+  \begin{warning}
+    Note hypotheses are always expressed in terms of population parameters
+  \end{warning}

 
   \subsection*{Null hypothesis \(H_0\)}
 
 
   \subsection*{Null hypothesis \(H_0\)}
 


@@ -218,23 +268,23 @@
 
   Significance level is denoted by \(\alpha\).
 
 
   Significance level is denoted by \(\alpha\).
 

-  If \(p<\alpha\), null hypothesis is \textbf{rejected} \\
-  If \(p>\alpha\), null hypothesis is \textbf{accepted}
+  \-\hspace{1em} If \(p<\alpha\), null hypothesis is \textbf{rejected} \\
+  \-\hspace{1em} If \(p>\alpha\), null hypothesis is \textbf{accepted}

 
   \subsection*{\(z\)-test}
 
   Hypothesis test for a mean of a sample drawn from a normally distributed population with a known standard deviation.
 
 
   \subsection*{\(z\)-test}
 
   Hypothesis test for a mean of a sample drawn from a normally distributed population with a known standard deviation.
 

-  \subsubsection*{\colorbox{cas}{\textbf{On CAS:}}}
-  
+  \begin{cas}

   Menu \(\rightarrow\) Statistics \(\rightarrow\) Calc \(\rightarrow\) Test. \\
   Select \textit{One-Sample Z-Test} and \textit{Variable}, then input:
   Menu \(\rightarrow\) Statistics \(\rightarrow\) Calc \(\rightarrow\) Test. \\
   Select \textit{One-Sample Z-Test} and \textit{Variable}, then input:

-  \begin{itemize}
-    \item \(\mu\) condition - same operator as \(H_1\)
-    \item \(\mu_0\) - expected sample mean (null hypothesis)
-    \item \(\sigma\) - standard deviation (null hypothesis)
-    \item \(\overline{x}\) - sample mean
-    \item \(n\) - sample size
-  \end{itemize}
+    \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=2cm, font=\normalfont]
+    \item[\(\mu\) cond:] same operator as \(H_1\)
+    \item[\(\mu_0\):] expected sample mean (null hypothesis)
+    \item[\(\sigma\):] standard deviation (null hypothesis)
+    \item[\(\overline{x}\):] sample mean
+    \item[\(n\):] sample size
+  \end{description}
+  \end{cas}

 
 \end{document}
 
 \end{document}