\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{NaeemahJ}
\pdfinfo{
  /Title (inferential-statistics.pdf)
  /Creator (Cheatography)
  /Author (NaeemahJ)
  /Subject (Inferential Statistics 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}{33A39C}
\definecolor{LightBackground}{HTML}{F2F9F8}
\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{Inferential Statistics Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{NaeemahJ} via \textcolor{DarkBackground}{\uline{cheatography.com/196521/cs/41490/}}}
\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}NaeemahJ \\
  \uline{cheatography.com/naeemahj} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 5th December, 2023.\\
   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}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Inferential Statistics Pathway}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701310449_Screenshot 2023-11-29 at 9.11.30 PM.png}}} \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}{Inferential Statistics}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Concept:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Making generalization about population parameter based on the sample statistic} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Examples} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Is there a difference in participation in local decision-makings between low-income and middle-income groups in New Jersey?} \tn 
% Row Count 7 (+ 4)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}What are reactions of New Brunswick residents to new investments of Rutgers University in development of College Avenue?} \tn 
% Row Count 10 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}What is the level of effectiveness of the Corona vaccine developed by Johnson and Johnson pharmaceutical company?} \tn 
% Row Count 13 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Primary Date: Sampling Methods}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{1) Probability Sampling}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\emph{Concept:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Sample is selected randomly (Equal Probability of Selection Method – EPSEM).} \tn 
% Row Count 4 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\emph{Methods:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Simple random / Systematic / Cluster / Stratified / Convenience} \tn 
% Row Count 7 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\emph{Application}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Diverse population / Generalization is required} \tn 
% Row Count 9 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{2) Non-Probability Sampling}}} \tn 
% Row Count 10 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\emph{Concept:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Sample selection is based on the subjective judgment of researcher.} \tn 
% Row Count 13 (+ 3)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\emph{Methods:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Judgement / Snowball /  Quota / Consecutive} \tn 
% Row Count 15 (+ 2)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\emph{Application}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Homogenous population / Pilot study} \tn 
% Row Count 17 (+ 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}{Probability Sampling Methods}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Simple Sampling}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Select a simple random sample through drawing or use of random number methods} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Systematic Sampling}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}1) Number all members of population sequentially. 2) From a starting point select every nth individual.} \tn 
% Row Count 7 (+ 4)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Cluster Sampling}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}1) Divide population into non-overlapping clusters. 2) Sample all in some clusters  (Ex. Sample all nurses in 5 hospitals in New Jersey).} \tn 
% Row Count 11 (+ 4)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Stratified Sampling}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}1) Divide population into non-overlapping clusters. 2) Sample some in clusters  (Ex. Sample some nurses in every hospitals in New Jersey).} \tn 
% Row Count 15 (+ 4)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Convenience Sampling}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Create a sample by using data from population members that are readily available.} \tn 
% Row Count 18 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Non-Probability Sampling Methods}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Judgement Sampling}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Select samples based on researcher's knowledge and if they fit to participate in the research - some subjects are more fit for the research compared to other individuals.} \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Snowball Sampling}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Once the researcher finds suitable subjects, he asks them for assistance to seek similar subjects to form a considerably good size sample - good or small population.} \tn 
% Row Count 10 (+ 5)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Quota Sampling}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Select equal or proportionate subjects depending on basis of the quota, which usually are age, gender, education, race, religion and socioeconomic status. (Ex. Sample size of 100, researcher can select 25 1st year students, 25 2nd year, 25 3rd year and 25 4th year students).} \tn 
% Row Count 17 (+ 7)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Consecutive Sampling}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Is very similar to convenience sampling except that it seeks to include ALL accessible subjects as part of the sample.} \tn 
% Row Count 21 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Data Biases}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Moderator / Interviewer bias }}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}The moderator's facial expressions, body language, tone, manner of dress, and style of language may introduce bias. Similarly, the moderator's age, social status, race, and gender can produce bias.} \tn 
% Row Count 6 (+ 6)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{BIased Questions}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A biased question influences respondents' answers. And the way you ask a question, or "vague wording" can bias a question.} \tn 
% Row Count 10 (+ 4)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Biased Answers }}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A biased answer is an untrue or partially true statement, like 1) Nonresponse in survey, when individuals can't or refuse to respond, 2) Truthfulness of response, 3) Faulty recall or not remember accurately, 4) Voluntary response: individuals with strong feelings about a subject are more likely than other to respond} \tn 
% Row Count 18 (+ 8)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Biased Samples }}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Poor screening and recruiting causes biased samples. Examples are biases of time and location} \tn 
% Row Count 21 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Biased Reporting }}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Experiences, beliefs, feelings, wishes, attitudes, culture, views, state of mind, reference, error, and personality can bias analysis and reporting.} \tn 
% Row Count 26 (+ 5)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
% Row Count 26 (+ 0)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Biased Questions }}} \tn 
% Row Count 27 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Estimation of Population Parameter}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Sample Statistics:} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Because of several constraints of collecting data from all individuals in population, including limitations of time, money, and labor, we usually depend on the sample information.} \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Point of Estimate:} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}The value of the parameters are unknown, but sample statistic is available, and can be used to estimate the population value} \tn 
% Row Count 9 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Characteristics of Sample Statistics: \newline Sample should represent the population value \newline Sampling techniques \newline Sample size}  \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}{Estimation of Population Value}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Concept:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Estimation of population parameter based on a sample statistic} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Example:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Rutgers Parking Authority collected data from a random sample of 220 Rutgers University students in order to estimate commuting time of all the university students.} \tn 
% Row Count 8 (+ 5)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}What percent of New Brunswick residents are aware of the advocacy planning efforts of the local governments in Middlesex County?} \tn 
% Row Count 11 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Study of 937 adults in NJ shows mean cholesterol level of 196 and standard deviation of 18 points. What can be concluded about the cholesterol level of all adults in NJ?} \tn 
% Row Count 15 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Estimation: Confidence Interval}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Concept of the Level of Confidence:}} & Confidence level reflects probability that interval of estimate presents actual value of population \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
{\bf{Structure:}} & Confidence Interval is a method of inferential statistics that helps us to use sample statistic to estimate population parameter. It is based on point of estimate and margin of error. \tn 
% Row Count 15 (+ 10)
% Row 2
\SetRowColor{LightBackground}
Confidence Interval = & Point of Estimate ± Margin of Error \tn 
% Row Count 17 (+ 2)
% Row 3
\SetRowColor{white}
{\bf{Translation of the Level of Confidence to z score}} & 𝑧 score in confidence interval formula presents level of confidence, such that area under normal curve between −𝑧 and +z is equal to level of confidence \tn 
% Row Count 25 (+ 8)
% Row 4
\SetRowColor{LightBackground}
{\bf{Requirements:}} & Sampling distribution is normal: ▪ Population distribution is normal ▪ Sample size is large (n\textgreater{}100) \tn 
% Row Count 31 (+ 6)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Estimation: Confidence Interval (cont)}}  \tn
% Row 5
\SetRowColor{LightBackground}
 & Population standard deviation (σ) is known, Sample is taken randomly \tn 
% Row Count 4 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Central Limit Theorem}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Concept:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Set of ideas about the relationship between sample and population (sampling distribution and normality, and difference)} \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Population mean equals to the mean of the sampling distribution} \tn 
% Row Count 6 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}If population distribution is normal, sampling distribution is also normal.} \tn 
% Row Count 8 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Even if the population distribution is not normal, distribution of sample means approaches normal distribution as sample size is increased, and sampling distribution is almost normal if sample size is large (n \textgreater{}100).} \tn 
% Row Count 13 (+ 5)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Sampling Error:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}is the difference between a sample statistics and corresponding population parameter.} \tn 
% Row Count 16 (+ 3)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Sampling error can not be determined} \tn 
% Row Count 17 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Sampling error most probably decreases as the sample size is increased.} \tn 
% Row Count 19 (+ 2)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Standard Error: }}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}measures the accuracy with which a sample distribution represents a population by using standard deviation.} \tn 
% Row Count 23 (+ 4)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}𝜎⁄√𝑛} \tn 
% Row Count 24 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Sample Size}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Concept:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}What is the minimum sample size required for study} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Examples:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A researcher wants to collect data from a random sample of Rutgers University students in order to estimate commuting time of all the university students. How many students he must study?} \tn 
% Row Count 8 (+ 5)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}What is the proper sample size if you want to check a planner's claim that more than 75\% of East Brunswick residents support increase of property tax to investment on development of renewable energy projects for the city?} \tn 
% Row Count 13 (+ 5)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}How large a sample must be in order to be 95\% sure about the outcome of our study?} \tn 
% Row Count 15 (+ 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}{Sample Size: Quantitative Data (Mean)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701733389_Screenshot 2023-12-04 at 6.42.39 PM.png}}} \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}{Sample Size: Qualitative Data (Proportion)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701733683_Screenshot 2023-12-04 at 6.47.18 PM.png}}} \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}{//}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \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}{Inferential Statistics: Hypothesis Testing}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Concept:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Making inference about population parameter based on a sample statistic} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Example:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}An officer at Rutgers Parking Authority collected data from a random sample of 220 Rutgers University students in order to check the idea that daily commuting time of Rutgers students is more than 45 minutes.} \tn 
% Row Count 9 (+ 6)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A researcher claims that less than one quarter of New Brunswick residents are aware of the advocacy planning efforts of the local governments in Middlesex County? (Qualitative)} \tn 
% Row Count 13 (+ 4)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Study of 937 adults in NJ shows mean cholesterol level of 196 and standard deviation of 18 points. Can we conclude that comparing to two year ago, the mean cholesterol level of NJ adults has increased?} \tn 
% Row Count 18 (+ 5)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{(Confidence Interval)} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Make an estimation about population parameter} \tn 
% Row Count 20 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{(Hypothesis Testing)} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Test validity of a claim about population parameter} \tn 
% Row Count 23 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Hypothesis Testing: Factors Affecting Decision}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Sampling Error}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Greater the gap between sample statistics and population parameter, greater is probability of rejecting the null hypothesis} \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Sampling Size}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Greater the sample size, greater is probability of rejecting the null hypothesis} \tn 
% Row Count 7 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{The Level of Significance}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Greater the level of significance, greater is probability of rejecting the null hypothesis} \tn 
% Row Count 10 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Hypothesis Testing: Steps}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701735074_Screenshot 2023-12-04 at 7.10.07 PM.png}}} \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}{Hypothesis Testing: One Population}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701780121_Screenshot 2023-12-05 at 7.37.40 AM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Concept: Research involves study of one population}  \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}{Hypothesis Testing: Two Populations}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Concept:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Making inference about comparison of two population parameters based on two sample statistics, one from each population} \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Example:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Does gender makes a difference in duration of commuting time to work?} \tn 
% Row Count 7 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Is Tylenol more popular than Advil?} \tn 
% Row Count 8 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Does GPA vary between undergraduate and graduate students?} \tn 
% Row Count 10 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A politician claims that when comparing democrats and republicans, a greater percentage of democrats support idea of dialogue among civilizations. Is this a valid claim?} \tn 
% Row Count 14 (+ 4)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Independent vs Dependent Populations}}} \tn 
% Row Count 15 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\emph{Independent Populations / Samples}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}The sample selected from one population is not related to the sample selected from the second population. // Changes within one population aren't related to changes within another population // Samples are randomly selected from these populations.} \tn 
% Row Count 22 (+ 7)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}With independent populations, we directly consider the differences in data points.} \tn 
% Row Count 24 (+ 2)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Example:} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Comparing drug A with drug B // Comparing degree of spread of coronavirus in NJ vs CA} \tn 
% Row Count 27 (+ 3)
% Row 9
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\emph{Dependen Populations / Samples (Paired or Matched)}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Each member of one sample corresponds to a member of the other sample. // Data Pairs occur naturally, most often with one data point occurring "before" and another data point occurring "after" an event.} \tn 
% Row Count 34 (+ 7)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Hypothesis Testing: Two Populations (cont)}}  \tn
% Row 10
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}With dependent populations, we pair the data points then consider the differences in data points.} \tn 
% Row Count 3 (+ 3)
% Row 11
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Example:} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Blood pressure of students before and after exam} \tn 
% Row Count 6 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Hypothesis Testing: Qualitative Data}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701737825_Screenshot 2023-12-04 at 7.55.53 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Concept: Making inference about population parameter based on a sample statistic for qualitative data (nominal and ordinal - proportion) \newline Ex: Are proportion of households headed by single parent in the lower-income neighborhoods significantly different from the general population? \newline Ex: Are the police arrest rates in Middlesex county significantly less than the statewide rate?}  \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}{Independent Populations: Qualitative Data (Z)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701789200_Screenshot 2023-12-05 at 10.11.10 AM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Requirements for qualitative data (Proportion, z): Independent populations \newline ▪ One sample is taken randomly from each population \newline ▪ To use a normal distribution, for each population: \newline ▪ Sample size is large (n \textgreater{} 100)}  \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}{Independent Populations: Quantitative Data (Z)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701782377_Screenshot 2023-12-05 at 8.18.31 AM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Requirements: Independent populations \newline ▪ One sample is taken randomly from each population \newline ▪ Sampling distribution is normal for each population: \newline ▪ Population distribution is normal \newline ▪ Sample size is large \newline ▪ Standard deviation of each population (σ) is known}  \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}{Independent Populations: Quantitative Data (T)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701788854_Screenshot 2023-12-05 at 10.06.17 AM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Requirements for quantitative data (Mean, t) : Independent populations \newline ▪ One sample is taken randomly from each population \newline ▪ Sampling distribution is normal for each population: \newline ▪ Population distribution is normal \newline ▪ Sample size is small \newline ▪ Standard deviation of two populations (σ) is unknown}  \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}{// (copy)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \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}{Confidence Interval- Quantitative (Z)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701729967_Screenshot 2023-12-04 at 5.44.45 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Quantitative (ie. Normal): Confidence interval for estimation of mean \newline ▪ Ex: What percentage of Rutgers University students study more than 25 hours per week?}  \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}{T Distribution}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701732535_Screenshot 2023-12-04 at 6.19.49 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.59264 cm} x{3.38436 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Confidence Interval: Quantitative Data (t)}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{T-Distribution }} & used when estimating the mean of a normally distributed population in situations where the sample size is small, and/or the population standard deviation is unknown. \tn 
% Row Count 7 (+ 7)
% Row 1
\SetRowColor{white}
 & The shape of distribution depends on the degrees of freedom (df = n-1). \tn 
% Row Count 10 (+ 3)
% Row 2
\SetRowColor{LightBackground}
 & As the degrees of freedom increase, the t distribution approaches the standard normal distribution. \tn 
% Row Count 14 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Confidence Interval: Qualitative Data (z)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701730285_Screenshot 2023-12-04 at 5.48.00 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Qualitative (ie.Binomial): Confidence interval for estimation of proportion \newline ▪ Ex: A researcher claims that at least 72\% of Americans support Green Solutions for sustainable urban renewal.}  \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}{// (copy) (copy)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \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}{Analysis of Variance (ANOVA)}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Applications:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Mean differences across several populations} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Potentials:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Works for comparison of three or more populations} \tn 
% Row Count 5 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Population with significant variation}}} \tn 
% Row Count 6 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\emph{Post Hoc Test:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}The alternative hypothesis is not specific – it only states that at least one of the population means differs from the others. To find which category is different and how much, use post hoc (or after the fact) techniques.} \tn 
% Row Count 12 (+ 6)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\emph{Series of two Populations t Test:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}To find the difference or critical group, you can use several two populations population analysis as needed (pick the important one, and then do the important one with another, and so on.} \tn 
% Row Count 17 (+ 5)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Different Types of ANOVA}}} \tn 
% Row Count 18 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{One-way ANOVA:}}}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}One-way ANOVA is used to test effects of a single independent variable (categorical) on a single dependent variable (numerical).} \tn 
% Row Count 22 (+ 4)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Example:} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Does income vary among African-Americans, Whites, Hispanic, and Other ethnic groups? This example includes only ONE dependent variable (income) and ONE independent variable (ethnicity).} \tn 
% Row Count 27 (+ 5)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{Two-way ANOVA:}}}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Two-way ANOVA is used to test effects of TWO independent variables on a single dependent variable} \tn 
% Row Count 31 (+ 4)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Analysis of Variance (ANOVA) (cont)}}  \tn
% Row 9
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Example:} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Does income vary in relation to ethnicity and personality?} \tn 
% Row Count 3 (+ 3)
% Row 10
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{Factorial ANOVA:}}}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Factorial ANOVA is used to test effects of SEVERAL (two or more) independent variables on a single dependent variable.} \tn 
% Row Count 7 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{ANOVA is appropriate for situations in which? we are comparing more than two populations}  \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}{ANOVA}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701796050_Screenshot 2023-12-05 at 11.52.54 AM.png}}} \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}{Hypothesis Testing: Analysis of Variance (ANOVA)}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Concept:} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Test of difference in the mean of {\bf{several populations}} (Analysis of Variance)} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}{\emph{Think of ANOVA as extension of t test for more than two populations}}} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Example:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Is there a difference among Protestants, Catholics and Jews in terms of number of children?} \tn 
% Row Count 8 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}How do Republicans, Democrats, and Independents vary in terms of income?} \tn 
% Row Count 10 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}How do older, middle-aged, and younger people vary in terms of supermarket shopping duration?} \tn 
% Row Count 12 (+ 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}{// (copy)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \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}{Hypothesis Testing: Chi Square}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701796390_Screenshot 2023-12-05 at 12.12.29 PM.png}}} \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}{Chi Square Formula}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/naeemahj_1701798895_Screenshot 2023-12-05 at 12.17.26 PM.png}}} \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}{Chi Square: Test of Independence}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Concept:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Test the relationship between 2 categories through their sub-categories} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Example:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}We want to know if gender and type of food people prefer are associated.} \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}We want to know if education level and political affiliation are associated.} \tn 
% Row Count 8 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Is type of sport students like associated with their ethnic background?} \tn 
% Row Count 10 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Application: }}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A Chi-Square ( 𝜒2) Test of Independence is used to determine existence of a significant association between two categorical variables.} \tn 
% Row Count 14 (+ 4)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
% Row Count 14 (+ 0)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Is type of sport students like associated with their ethnic background?} \tn 
% Row Count 16 (+ 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}{Chi Square: Test of Goodness-of-Fit}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Concept:}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Test validity of a particular distribution of population subcategories (how well sample data fit a distribution from a population with a normal distribution).} \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Example}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A professor claims that in metropolitan areas, 19\% of college students are Black, 42\% are White, 16\% are Hispanic, and the rest have a different ethnic background. Is this a valid claim?} \tn 
% Row Count 10 (+ 5)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}\seqsplit{Isthisvalidtoassumethatlocalgymshavetheirhighest} attendance on Mondays, Tuesdays and Saturdays, average attendance on Wednesdays and Thursdays, and lowest attendance on Fridays and Sundays.} \tn 
% Row Count 15 (+ 5)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}A researcher claims that season makes no difference in number of gun violations in large urban centers.} \tn 
% Row Count 18 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Applications}}} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Study pattern of distribution of subgroups} \tn 
% Row Count 20 (+ 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}{// (copy)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \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}{// (copy)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}