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wk7/week7.tex
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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}
\subsection{Visualisation} \label{subsec:visualisation}
\subsubsection{Effect of Height} \label{subsubsec:vheight}
\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.
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.
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.
\subsubsection{Effect of Handedness} \label{subsubsec:vhand}
We use distribution plots from Seaborn to visualise the distribution of points earned by left-handed and right-handed players overlapped on the same plot. The visualisation uses a kernel density estimate of the probability density function derived from the sample provided. We also plot separate distributions for male and female players in case there are any noticeable differences between genders.
\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}
\caption{Correlation matrix of the numerical values in the dataset}
\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}
\caption{Scatter plot of points against height with an outlier}
\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}
\caption{Scatter plot of points against height with the outlier removed}
\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}
\caption{Scatter plot of height against rank}
\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}
\caption{Distribution plots of points separated by handedness}
\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}
\caption{Distribution plots of points separated by handedness for males and distribution plots of points separated by handedness for females}
\end{figure}
\subsection{Statistical Tests} \label{subsec:stattests}
In testing the first hypothesis, we perform T-tests to analyse the effect of height on the points earned by players. Two T-tests are performed; one for each gender. Each gender of players are separated into two groups; a group of players that scored above the mean number of points and a group of players that scored below the mean number of points and these groups are compared in the T-tests. Later we perform a $\chi^2$ test on the groups together.
To test the second hypothesis, we use a T-test to measure the effect of handedness and a $\chi^2$ test to measure the difference between the expected values and the observed values and garner a probability that the sample belongs to the $\chi^2$ distribution.
\section{Results} \label{sec:results}
\subsection{The advantage of height}