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}
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))}%
+ \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}
}
-{\begin{center} \begin{tikzpicture}
+\begin{tikzpicture}
\pgfplotsset{set layers}
\begin{axis}[every axis plot post/.append style={
mark=none,domain=-3:3,samples=50,smooth},
every axis x label/.style={at={(axis description cs:1,-0.25)},anchor=south west}]
\addplot {gauss(0,0.75)};
\end{axis}
-\end{tikzpicture}\end{center}}
+\end{tikzpicture}
\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}