\documentclass[10pt,a4paper]{article} % Packages \usepackage{fancyhdr} % For header and footer \usepackage{multicol} % Allows multicols in tables \usepackage{tabularx} % Intelligent column widths \usepackage{tabulary} % Used in header and footer \usepackage{hhline} % Border under tables \usepackage{graphicx} % For images \usepackage{xcolor} % For hex colours %\usepackage[utf8x]{inputenc} % For unicode character support \usepackage[T1]{fontenc} % Without this we get weird character replacements \usepackage{colortbl} % For coloured tables \usepackage{setspace} % For line height \usepackage{lastpage} % Needed for total page number \usepackage{seqsplit} % Splits long words. %\usepackage{opensans} % Can't make this work so far. Shame. Would be lovely. \usepackage[normalem]{ulem} % For underlining links % Most of the following are not required for the majority % of cheat sheets but are needed for some symbol support. \usepackage{amsmath} % Symbols \usepackage{MnSymbol} % Symbols \usepackage{wasysym} % Symbols %\usepackage[english,german,french,spanish,italian]{babel} % Languages % Document Info \author{Natalie Moore (NatalieMoore)} \pdfinfo{ /Title (hypothesis-testing.pdf) /Creator (Cheatography) /Author (Natalie Moore (NatalieMoore)) /Subject (Hypothesis Testing Cheat Sheet) } % Lengths and widths \addtolength{\textwidth}{6cm} \addtolength{\textheight}{-1cm} \addtolength{\hoffset}{-3cm} \addtolength{\voffset}{-2cm} \setlength{\tabcolsep}{0.2cm} % Space between columns \setlength{\headsep}{-12pt} % Reduce space between header and content \setlength{\headheight}{85pt} % If less, LaTeX automatically increases it \renewcommand{\footrulewidth}{0pt} % Remove footer line \renewcommand{\headrulewidth}{0pt} % Remove header line \renewcommand{\seqinsert}{\ifmmode\allowbreak\else\-\fi} % Hyphens in seqsplit % This two commands together give roughly % the right line height in the tables \renewcommand{\arraystretch}{1.3} \onehalfspacing % Commands \newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour \newcommand{\mymulticolumn}[3]{\multicolumn{#1}{>{\columncolor{\RowColorName}}#2}{#3}} % For coloured multi-cols \newcolumntype{x}[1]{>{\raggedright}p{#1}} % New column types for ragged-right paragraph columns \newcommand{\tn}{\tabularnewline} % Required as custom column type in use % Font and Colours \definecolor{HeadBackground}{HTML}{333333} \definecolor{FootBackground}{HTML}{666666} \definecolor{TextColor}{HTML}{333333} \definecolor{DarkBackground}{HTML}{5DA389} \definecolor{LightBackground}{HTML}{F4F9F7} \renewcommand{\familydefault}{\sfdefault} \color{TextColor} % Header and Footer \pagestyle{fancy} \fancyhead{} % Set header to blank \fancyfoot{} % Set footer to blank \fancyhead[L]{ \noindent \begin{multicols}{3} \begin{tabulary}{5.8cm}{C} \SetRowColor{DarkBackground} \vspace{-7pt} {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}} } \end{tabulary} \columnbreak \begin{tabulary}{11cm}{L} \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Hypothesis Testing Cheat Sheet}}}} \\ \normalsize{by \textcolor{DarkBackground}{Natalie Moore (NatalieMoore)} via \textcolor{DarkBackground}{\uline{cheatography.com/19119/cs/6012/}}} \end{tabulary} \end{multicols}} \fancyfoot[L]{ \footnotesize \noindent \begin{multicols}{3} \begin{tabulary}{5.8cm}{LL} \SetRowColor{FootBackground} \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}} \\ \vspace{-2pt}Natalie Moore (NatalieMoore) \\ \uline{cheatography.com/nataliemoore} \\ \uline{\seqsplit{www}.jchmedia.com/} \end{tabulary} \vfill \columnbreak \begin{tabulary}{5.8cm}{L} \SetRowColor{FootBackground} \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}} \\ \vspace{-2pt}Published 19th March, 2017.\\ Updated 12th May, 2016.\\ Page {\thepage} of \pageref{LastPage}. \end{tabulary} \vfill \columnbreak \begin{tabulary}{5.8cm}{L} \SetRowColor{FootBackground} \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}} \\ \SetRowColor{white} \vspace{-5pt} %\includegraphics[width=48px,height=48px]{dave.jpeg} Measure your website readability!\\ www.readability-score.com \end{tabulary} \end{multicols}} \begin{document} \raggedright \raggedcolumns % Set font size to small. Switch to any value % from this page to resize cheat sheet text: % www.emerson.emory.edu/services/latex/latex_169.html \footnotesize % Small font. \begin{multicols*}{3} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Statistical inference}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Draw conclusions from a set of data} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Put a probability on whether a conclusion is correct 'beyond reasonable doubt'} \tn % Row Count 3 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{The major question to answer is whether a difference between samples, or between a sample and a population, has occurred simply as a result of natural variation or because of a real difference between the two} \tn % Row Count 8 (+ 5) \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Two-tailed or one-tailed}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{The alternative hypothesis may be classified as two-tailed or one-tailed} \tn % Row Count 2 (+ 2) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Two-tailed test}}} \tn % Row Count 3 (+ 1) % Row 2 \SetRowColor{LightBackground} - & is a two-sided alternative \tn % Row Count 4 (+ 1) % Row 3 \SetRowColor{white} - & we do the test with no preconceived notion that the true value of μ is either above or below the hypothesised value of μ 0 \tn % Row Count 8 (+ 4) % Row 4 \SetRowColor{LightBackground} - & the alternative hypothesis is written: H1: µ =/= µo \tn % Row Count 10 (+ 2) % Row 5 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{One-tailed test}}} \tn % Row Count 11 (+ 1) % Row 6 \SetRowColor{LightBackground} - & one-sided alternative \tn % Row Count 12 (+ 1) % Row 7 \SetRowColor{white} - & do the test with a strong conviction that, if H0 is not true, it is clear that m is either grater than µ0 or less than µ0 \tn % Row Count 16 (+ 4) % Row 8 \SetRowColor{LightBackground} - & E.g. the alternative hypothesis is written as: H1: µ \textgreater{} µo \tn % Row Count 18 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Decision-making process steps}} \tn % Row 0 \SetRowColor{LightBackground} 1. & Collecting the data \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} 2. & Summarising the data \tn % Row Count 2 (+ 1) % Row 2 \SetRowColor{LightBackground} 3. & Setting up a hypothesis (i.e. a claim or theory), which is to be tested \tn % Row Count 4 (+ 2) % Row 3 \SetRowColor{white} 4. & Calculating the probability of obtaining a sample such as the one we have if the hypothesis is true \tn % Row Count 7 (+ 3) % Row 4 \SetRowColor{LightBackground} 5. & Either accepting or rejecting the hypothesis \tn % Row Count 9 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Significance level}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{After the appropriate hypotheses have been formulated, we must decide upon the significance level (or α -level) of the test} \tn % Row Count 3 (+ 3) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{most common significance level used is 0.05, commonly written as α = 0.05} \tn % Row Count 5 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{A 5\% significance level says in effect that an event has occurred that occurs less than 5\% of the time is considered unusual} \tn % Row Count 8 (+ 3) \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{One-sample z-test}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Deals with the case of a single sample being chosen from a population and the question of whether that particular sample might be consistent with the rest of the population} \tn % Row Count 4 (+ 4) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Construct a test statistic according to a particular formula} \tn % Row Count 6 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Information required in calculation} \tn % Row Count 7 (+ 1) % Row 3 \SetRowColor{white} - & the size (n) of the sample \tn % Row Count 8 (+ 1) % Row 4 \SetRowColor{LightBackground} - & the mean of the sample \tn % Row Count 9 (+ 1) % Row 5 \SetRowColor{white} - & the standard deviation (s) of the sample \tn % Row Count 11 (+ 2) % Row 6 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Other information of interest might include:} \tn % Row Count 12 (+ 1) % Row 7 \SetRowColor{white} - & Does the population have a normal distribution? \tn % Row Count 14 (+ 2) % Row 8 \SetRowColor{LightBackground} - & Is the population's standard deviation known? \tn % Row Count 16 (+ 2) % Row 9 \SetRowColor{white} - & Is the sample size (n) large? (25+) \tn % Row Count 17 (+ 1) % Row 10 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{There are different cases for the one-sample z-test statistic} \tn % Row Count 19 (+ 2) % Row 11 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Case I }}} \tn % Row Count 20 (+ 1) % Row 12 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{the population has a normal distribution and} \tn % Row Count 21 (+ 1) % Row 13 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{the population standard deviation, s, is known} \tn % Row Count 22 (+ 1) % Row 14 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Case II }}} \tn % Row Count 23 (+ 1) % Row 15 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{the population has any distribution} \tn % Row Count 24 (+ 1) % Row 16 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{the sample size, n, is large (i.e. at least 25), and} \tn % Row Count 26 (+ 2) % Row 17 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{the value of population standard deviation is known} \tn % Row Count 28 (+ 2) % Row 18 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\emph{In both these cases we can use a z-test statistic formula (a)}}} \tn % Row Count 30 (+ 2) \end{tabularx} \par\addvspace{1.3em} \vfill \columnbreak \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{One-sample z-test (cont)}} \tn % Row 19 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Case III }}} \tn % Row Count 1 (+ 1) % Row 20 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{the population has any distribution} \tn % Row Count 2 (+ 1) % Row 21 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{the sample size, n, is large (i.e. at least 25), and} \tn % Row Count 4 (+ 2) % Row 22 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{the value of population standard devation is unknown (however, since n is large, the value of population standard devation is approximated by the sample standard deviation, s)} \tn % Row Count 8 (+ 4) % Row 23 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\emph{In this case we can use a z-test statistic formula (b)}}} \tn % Row Count 10 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Set up your Hypothesis}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Null Hypothesis}}} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Part of formulation of an hypothesis} \tn % Row Count 2 (+ 1) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Statement that nothing unusual has occurred} \tn % Row Count 3 (+ 1) % Row 3 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{The notation is Ho} \tn % Row Count 4 (+ 1) % Row 4 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Alternative hypothesis }}} \tn % Row Count 5 (+ 1) % Row 5 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{States that something unusual has occurred} \tn % Row Count 6 (+ 1) % Row 6 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{The notation is H1 or HA} \tn % Row Count 7 (+ 1) % Row 7 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Together}} they may be written in the form: Ho: (statement) v. H1(alternative statement)} \tn % Row Count 9 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Conclusion errors}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Two possible errors in making a conclusion about a null hypothesis} \tn % Row Count 2 (+ 2) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Type I errors occur when you reject H0 (i.e. conclude that it is false) when H0 is really true.} \tn % Row Count 4 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Type II errors occur when you accept H0 (i.e. conclude that it is true) when H0 is really false.} \tn % Row Count 6 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{z-test statistic formula (a)}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/nataliemoore_1446994473_2015-11-09 00_53_45-Photos.jpg}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{z-test statistic formula (b)}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/nataliemoore_1446994751_2015-11-09 00_57_27-Photos.jpg}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} % That's all folks \end{multicols*} \end{document}