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% Document Info
\author{mmmmy}
\pdfinfo{
  /Title (hypothesis-testing-cheatsheet.pdf)
  /Creator (Cheatography)
  /Author (mmmmy)
  /Subject (Hypothesis testing cheatsheet Cheat Sheet)
}

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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Hypothesis testing cheatsheet Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{mmmmy} via \textcolor{DarkBackground}{\uline{cheatography.com/132869/cs/26952/}}}
\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}mmmmy \\
  \uline{cheatography.com/mmmmy} \\
  \end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 10th March, 2021.\\
   Updated 10th March, 2021.\\
   Page {\thepage} of \pageref{LastPage}.
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




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

\begin{tabularx}{8.4cm}{x{4 cm} x{4 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{{\bf{ Introduction }}}}  \tn
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Statistical hypothesis & Statistical hypothesis testing \tn 
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a hypothesis that is testable on the basis of observed data modeled as the realized values taken by a collection of random variables & a statistical way of testing the assumption regarding a popular parameter \tn 
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\mymulticolumn{2}{x{8.4cm}}{\{\{link="https://en.wikipedia.org/wiki/Statistical\_hypothesis\_testing"\}\}}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{{\bf{steps of formulating a hypothesis}}}}  \tn
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\mymulticolumn{1}{x{8.4cm}}{1. state the two hypothesis: {\bf{Null hypothesis}} and {\bf{Alternative hypothesis }}} \tn 
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\mymulticolumn{1}{x{8.4cm}}{2. set the {\bf{significance levels}} usually α  = 0.05} \tn 
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\mymulticolumn{1}{x{8.4cm}}{3. carrying out the hypothesis testing and calculate the test statistics and corresponding {\bf{P-value}}} \tn 
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\mymulticolumn{1}{x{8.4cm}}{4. compare P-value with significance levels and then decide to accept or reject null hypothesis} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{1.08 cm} x{2.664 cm} x{1.512 cm} x{1.944 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{8.4cm}}{\bf\textcolor{white}{Errors in Testing}}  \tn
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{\bf{Error Types}} & {\bf{Description}} & {\bf{denotation}} & {\bf{correct inference}} \tn 
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Type I error & {\bf{Reject}} null when null is true & α = P(Type I error) & 1 - α \seqsplit{(significance} level) \tn 
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Type II error & {\bf{Not}} reject null when null is false & β = P(Type II error) & 1 - β (= power) \tn 
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\begin{tabularx}{8.4cm}{x{2.96 cm} x{5.04 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Chi-Square Test}}  \tn
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Types & Description \tn 
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Test for independence & tests for the independence of two categorical variables \tn 
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Homogeneity of Variance & test if more than two subgroups of a population share the same multivariate distribution \tn 
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goodness of fit & whether a multinomial model for the population distribution (P1,....Pm) fits our data \tn 
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\mymulticolumn{2}{x{8.4cm}}{Test for independence and homogeneity of variance share the same test statistics and degree of freedoms by different design of experiment \newline  \newline {\bf{Assumptions}} \newline 1. one or two categorical variables \newline  2. independent observations  \newline 3. outcomes mutually exclusive  \newline 4. large n and no more than 20\% of expected counts \textless{} 5}  \tn 
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\end{tabularx}
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\begin{tabularx}{8.4cm}{x{2.08 cm} x{5.92 cm} }
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{F-test}}  \tn
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Anova Analysis & comparing the means of two or more continuous populations \tn 
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One-way layout & A test that allows one to make comparisons between the means of two or more groups of data. \tn 
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two-way layout & A test that allows one to make comparisons between the means of two or more groups of data, where two independent variables are considered. \tn 
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\mymulticolumn{2}{x{8.4cm}}{Assumptions about data: \newline 1. each data y is normally distributed \newline 2. the variance of each treatment group is same \newline 3. all observations are independent}  \tn 
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{{\bf{T-test}}}}  \tn
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Types & Hypothesis \tn 
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{\bf{Two Sample T-test}} & If two independent groups have different mean \tn 
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{\bf{Paired T-test}} & if one groups have different means at different times \tn 
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{\bf{One Sample T-test}} & mean of a single group against a known mean \tn 
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\mymulticolumn{2}{x{8.4cm}}{Assumptions about data  \newline 1. independent \newline 2. normally distributed \newline 3. have a similar amount of variance within each group being compared}  \tn 
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\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{{\bf{Test Statistics formula -{}- T-test}}}}  \tn
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\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{One sample T-test}}  \tn
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\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/mmmmy_1615278339_Screen Shot 2021-03-09 at 4.19.50 PM.png}}} \tn 
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\mymulticolumn{1}{x{8.4cm}}{where \newline  m = the mean of sample  \newline  s = standard deviation of sample  \newline degree of freedom = n - 1}  \tn 
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\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Paired T-test statistics}}  \tn
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\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/mmmmy_1615278125_Screen Shot 2021-03-09 at 4.20.26 PM.png}}} \tn 
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\mymulticolumn{1}{x{8.4cm}}{where \newline m = the mean of differences between two paired sets of data  \newline n = size of differences \newline s = the standard deviation of differences between two paired sets of data \newline degree of freedom = n - 1}  \tn 
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\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Independent two-sample T-test statistics}}  \tn
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\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/mmmmy_1615278253_Screen Shot 2021-03-09 at 4.22.46 PM.png}}} \tn 
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\mymulticolumn{1}{x{8.4cm}}{where \newline m =  the means of group A and B respectively  \newline n =  the sizes of group A and B respectively \newline degrees of freedom = nA + nB - 2 (given two samples have the same variance)}  \tn 
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\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{{\bf{Test statistics  -{}-Chi-square test}}}}  \tn
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\mymulticolumn{1}{x{8.4cm}}{} \tn 
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\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Test of independence and Homogeneity of variance}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/mmmmy_1615283242_Screen Shot 2021-03-09 at 5.41.15 PM.png}}} \tn 
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\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{where \newline  Er,c = (Nr {\emph{ Nc)/n \newline df = (r - 1) }} (c - 1) \newline c = column number \newline r = row number}  \tn 
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\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Goodness of fit test}}  \tn
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\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/mmmmy_1615280777_Screen Shot 2021-03-09 at 5.04.40 PM.png}}} \tn 
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\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{where: \newline O = observed value of data \newline E = expected value of data  \newline k = dimension of parameter  \newline df = n -1 - k}  \tn 
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\end{tabularx}
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\begin{tabularx}{8.4cm}{p{0.76 cm} x{2.432 cm} x{4.408 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Carrying out one-way anova test}}  \tn
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SST & total variance & sum(Yij - overall mean of Y)\textasciicircum{}2\textasciicircum{} \tn 
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SSW & intra-group variance & sum(mean of each observations across different treatments - mean of each treatment)\textasciicircum{}2\textasciicircum{} \tn 
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SSB & inter-group variance & sum(mean of each treatments - overall mean of Y)\textasciicircum{}2\textasciicircum{} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}---}
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{Null hypothesis: the differentiated effect in each treatment group is 0 \newline Alternative hypothesis: not all differentiated effect is 0 \newline  \newline SST = SSW + SSB \newline  \newline test statistics: \newline  \newline Fi-1,i(j-1) = SSB/(I-1)/SSW/I(J-1) \newline  \newline where  \newline I = number of different treatments \newline J = number of observations within each treatment}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
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% That's all folks
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