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\begin{document}
	\title{Week 7 - Evidence and experiments}
	\author{
		Jai Bheeman \and Kelvin Davis \and Jip J. Dekker \and Nelson Frew \and Tony
		Silvestere
	}
	\maketitle

	\section{Introduction} \label{sec:introduction}

	\section{Method} \label{sec:method}

    We are testing two hypotheses. The first hypothesis that we test is that tall players have an advantage over smaller players. The second hypothesis that we test is that left-handed players have an advantage over right-handed players. To build an intuition of how the data behaves with respect to the hypotheses we are testing, we created visual representations using tools from the Matplotlib, and Seaborn libraries and then we perform statistical tests to measure these effects.

    \subsection{Visualisation} \label{subsec:visualisation}

		\subsubsection{Effect of Height} \label{subsubsec:vheight}

			We started by performing a scatter plot of points earned by players with respect to their heights, to which we were surprised to find a player recorded to approximately 18m tall. This, we found to be somewhat contradictory to the currently held record of 2.72m. Removing this outlier, we can see a sufficient spread in height, points and ranking. We can also see slight discrepancy in height between males and females, and because of this, we perform separate statistical tests on males and females as to remove the effect of the gender. We plot both points with respect to height and height with respect to ranking.

			The plot of height with respect to ranking does not show an explicit relationship between the two variables, however we aim to test this relation in the Results Section.

			\begin{figure}[ht]
				\centering
				\label{fig:distr}
				\includegraphics[width=\textwidth]{correlation.png}
				\caption{Distribution of each of the variables recorded in the data, as a function of the distance from an event}
			\end{figure}

			\begin{figure}[ht]
				\centering
				\label{fig:distr}
				\includegraphics[width=\textwidth]{outlier.png}
				\caption{Distribution of each of the variables recorded in the data, as a function of the distance from an event}
			\end{figure}

			\begin{figure}[ht]
				\centering
				\label{fig:distr}
				\includegraphics[width=\textwidth]{pointheight.png}
				\caption{Distribution of each of the variables recorded in the data, as a function of the distance from an event}
			\end{figure}

			\begin{figure}[ht]
				\centering
				\label{fig:distr}
				\includegraphics[width=\textwidth]{heightrank.png}
				\caption{Distribution of each of the variables recorded in the data, as a function of the distance from an event}
			\end{figure}

			\begin{figure}[ht]
				\centering
				\label{fig:distr}
				\includegraphics[width=\textwidth]{handdistr.png}
				\caption{Distribution of each of the variables recorded in the data, as a function of the distance from an event}
			\end{figure}

			\begin{figure}[ht]
				\centering
				\label{fig:distr}
				\includegraphics[width=\textwidth]{handdistr_gender.png}
				\caption{Distribution of each of the variables recorded in the data, as a function of the distance from an event}
			\end{figure}

	\section{Results} \label{sec:results}

	\subsection{The advantage of height}

	\begin{table}[ht]
		\centering
		\label{tab:chi-height}
		\begin{tabular}{|l|r|r|r|r|}
			\hline
												& \textbf{M: 168 - 188} & \textbf{M: 189 - 210} & \textbf{F: 155 - 171} & \textbf{F: 172 - 189} \\ \hline
		\textbf{1 - 99}    & 67 / 73               & 32 / 26               & 38 / 42               & 60 / 55               \\
		\textbf{100 - 199} & 69 / 72               & 30 / 26               & 31 / 27               & 32 / 36               \\
		\textbf{200 - 299} & 75 / 68               & 17 / 25               & 18 / 17               & 22 / 23               \\
		\textbf{300 - 399} & 61 / 60               & 21 / 23               & 11 /12                & 17 / 16               \\
		\textbf{400 - 499} & 59 / 60               & 22 / 22               & 7 / 6
		& 7 / 8 \\
		\hline
		\end{tabular}
		\caption{Observed / Expected values used for the $\chi^2$-test. The groups are divided by their rank (vertical) and, per gender, their height (horizontal).}
	\end{table}

	$$
		\chi^2 \approx 7.697606186049128
	$$

	$$
		df = (5-1)(4-1) = 12
	$$

	$$
		\chi^2(7.69\dots,12) \approx 0.8082925814979871
	$$


	\textbf {t-test men:} T score: 1.711723, P score: 0.043815

	\textbf {t-test women:} T score: 1.860241, P score: 0.032030


	\subsection{The advantage of left-handedness}

	\begin{table}[ht]
		\centering
		\label{tab:chi-hand}
		\begin{tabular}{|l|l|l|l|l|l|}
		\hline
								& \textbf{1 - 99} & \textbf{100 - 199} & \textbf{200 - 299} & \textbf{300 - 399} & \textbf{400 - 499} \\
		\hline
		\textbf{L} & 22 / 21         & 23 / 18            & 17 / 15            & 6  / 12            & 8 / 10             \\
		\textbf{R} & 174 / 177       & 139 / 144          & 117 / 119          &
		105 / 98           & 88 / 86 \\
		\hline
		\end{tabular}
		\caption{Observed / Expected values used for the $\chi^2$-test. The groups are divided by which hand they use (vertical) and their rank (horizontal).}
	\end{table}


	$$
		\chi^2 \approx 6.467312944404331
	$$

	$$
		df = (2-1)(5-1) = 4
	$$

	$$
		\chi^2(6.46\dots,4) \approx 0.1668616190847413
	$$

	\textbf {t-test:} T score: 0.451694, P score: 0.325815


	\section{Discussion} \label{sec:discussion}

\end{document}