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\fancyhead[LO,LE]{Unit 4 Specialist --- Statistics}
\fancyhead[CO,CE]{Andrew Lorimer}

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\begin{document}

  \title{Statistics}
  \author{}
  \date{}
  \maketitle

  \section{Linear combinations of random variables}

  \subsection*{Continuous random variables}

  A continuous random variable \(X\) has a pdf \(f\) such that:

  \begin{enumerate}
    \item \(f(x) \ge 0 \forall x \)
    \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
  \end{enumerate}

  \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]

  \subsubsection*{Linear functions \(X \rightarrow aX+b\)}

  \begin{align*}
    \Pr(Y \le y) &= \Pr(aX+b \le y) \\
    &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
    &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx
  \end{align*}

  \begin{align*}
    \textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\
    \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
  \end{align*}

  \subsection*{Linear combination of two random variables}

  \begin{align*}
    \textbf{Mean:} && \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
    \textbf{Variance:} && \operatorname{Var}(aX+bY) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) \tag{if \(X\) and \(Y\) are independent}\\
  \end{align*}

  \section{Sample mean}

  Approximation of the \textbf{population mean} determined experimentally.

  \[ \overline{x} = \dfrac{\Sigma x}{n} \]

  where \(n\) is the size of the sample (number of sample points) and \(x\) is the value of a sample point

  \begin{tcolorbox}[colframe=cas!75!black, title=On CAS]

  \begin{enumerate}
    \item Spreadsheet
    \item In cell A1: \verb;mean(randNorm(sd, mean, sample size));
    \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
    \item Input range as A1:An where \(n\) is the number of samples
    \item Graph \(\rightarrow\) Histogram
  \end{enumerate}
  \end{tcolorbox}

  \subsubsection*{Sample size of \(n\)}

  \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]

  Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).

  For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)

  \begin{tcolorbox}[colframe=cas!75!black, title=On CAS]
  
    \begin{itemize}
      \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left
      \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
    \end{itemize}
  \end{tcolorbox}
  
  \section{Normal distributions}

  mean = mode = median

  \[ Z = \frac{X - \mu}{\sigma} \]

  Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\)
\pgfmathdeclarefunction{gauss}{2}{%
  \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}
}

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  mark=none,domain=-3:3,samples=50,smooth}, 
  axis x line=bottom, 
  axis y line=left,
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  ytick={0.55},
  yticklabels={\(\frac{1}{\sigma \sqrt{2\pi}}\)}, 
  xtick={-2,-1,0,1,2},
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  ylabel={\(\Pr(X=x)\)}]
  \addplot {gauss(0,0.75)};
\end{axis}
\begin{axis}[every axis plot post/.append style={
  mark=none,domain=-3:3,samples=50,smooth}, 
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  enlargelimits=upper,
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  \addplot {gauss(0,0.75)};
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  \section{Central limit theorem}

  If \(X\) is randomly distributed with mean \(\mu\) and sd \(\sigma\), then with an adequate sample size \(n\) the distribution of the sample mean \(\overline{X}\) is approximately normal with mean \(E(\overline{X})\) and \(\operatorname{sd}(\overline{X}) = \frac{\sigma}{\sqrt{n}}\).

  \section{Confidence intervals}

  \begin{itemize}
    \item \textbf{Point estimate:} single-valued estimate of the population mean from the value of the sample mean \(\overline{x}\)
    \item \textbf{Interval estimate:} confidence interval for population mean \(\mu\)
  \end{itemize}

  \subsection{95\% confidence interval}

  The 95\% confidence interval for a population mean \(\mu\) is given by

  \[ \overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \]

  where: \\
  \(\overline{x}\) is the sample mean \\
  \(\sigma\) is the population sd \\
  \(n\) is the sample size from which \(\overline{x}\) was calculated

  Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.

  \colorbox{cas}{\textbf{On CAS}}

  Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
  Set Type = One-Sample Z Int, Variable

  \subsection*{Interpretation of confidence intervals}

  95\% confidence interval \(\implies\) 95\% of samples will contain population mean \(\mu\).

  \subsection*{Margin of error}

  For 95\% confidence interval for \(\mu\), margin of error \(M\) is:

  \begin{align*}
    M &= 1.96 \times \dfrac{\sigma}{\sqrt{n}} \\
    \implies n &= \left( \dfrac{1.96 \sigma}{M} \right)^2
  \end{align*}

  \subsection*{General case}

  A confidence interval of \(C\)\% for a mean \(\mu\)  s given by

  \[ x \in \left( \overline{x} \pm k \dfrac{\sigma}{\sqrt{n}} \right) \quad \text{ where } k \text{ is such that } \Pr(-k < Z < k) = \frac{C}{100} \]

  \subsection*{Confidence interval for multiple trials}

  For a set of \(n\) confidence intervals (samples), there is \(0.95^n\) chance that all \(n\) intervals contain the population mean \(\mu\).

  \section{Hypothesis testing}

  Note hypotheses are always expressed in terms of population parameters

  \subsection*{Null hypothesis \(H_0\)}

  Sample drawn from population has same mean as control population, and any difference can be explained by sample variations.

  \subsection*{Alternative hypothesis \(H_1\)}

  Amount of variation from control is significant, despite standard sample variations.

  \subsection*{\(p\)-value}

  Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true.

  % table of p-values for strength of evidence

  \subsection*{Distribution of sample mean}

  If \(X \sim \operatorname{N}(\mu, \sigma)\), then distribution of sample mean \(\overline{X}\) is also normal with \(\overline{X} \sim \operatorname{N}(\mu, \frac{\sigma}{\sqrt{n}}\).

  \subsection*{Statistical significance}

  Significance level is denoted by \(\alpha\).

  If \(p<\alpha\), null hypothesis is \textbf{rejected} \\
  If \(p>\alpha\), null hypothesis is \textbf{accepted}

  \subsection*{\(z\)-test}

  Hypothesis test for a mean of a sample drawn from a normally distributed population with a known standard deviation.

  \subsubsection*{\colorbox{cas}{\textbf{On CAS:}}}
  
  Menu \(\rightarrow\) Statistics \(\rightarrow\) Calc \(\rightarrow\) Test. \\
  Select \textit{One-Sample Z-Test} and \textit{Variable}, then input:
  \begin{itemize}
    \item \(\mu\) condition - same operator as \(H_1\)
    \item \(\mu_0\) - expected sample mean (null hypothesis)
    \item \(\sigma\) - standard deviation (null hypothesis)
    \item \(\overline{x}\) - sample mean
    \item \(n\) - sample size
  \end{itemize}

\end{document}