From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Thu, 29 Aug 2019 12:45:38 +0000 (+1000)
Subject: [spec] start hypothesis testing
X-Git-Tag: yr12~50
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/d63c834106f09ab25fb0948582acf2031edf4d90

[spec] start hypothesis testing
---

diff --git a/spec/statistics.pdf b/spec/statistics.pdf
index cd372e0..60f093c 100644
Binary files a/spec/statistics.pdf and b/spec/statistics.pdf differ
diff --git a/spec/statistics.tex b/spec/statistics.tex
index ee414c6..4c89dd5 100644
--- a/spec/statistics.tex
+++ b/spec/statistics.tex
@@ -103,10 +103,10 @@
 
   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}, 
@@ -138,7 +138,7 @@
   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}
 
@@ -151,12 +151,13 @@
     \item \textbf{Interval estimate:} confidence interval for population mean \(\mu\)
   \end{itemize}
 
-  \subsection*{95% confidence interval}
+  \subsection{95\% confidence interval}
 
-  \[ \left( \overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \]
+  The 95\% confidence interval for a population mean \(\mu\) is given by
+
+  \[ \overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \]
 
   where: \\
-  \(\mu\) is the population mean (unknown) \\
   \(\overline{x}\) is the sample mean \\
   \(\sigma\) is the population sd \\
   \(n\) is the sample size from which \(\overline{x}\) was calculated
@@ -170,11 +171,11 @@
 
   \subsection*{Interpretation of confidence intervals}
 
-  95% confidence interval \(\implies\) 95% of samples will contain population mean \(\mu\).
+  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:
+  For 95\% confidence interval for \(\mu\), margin of error \(M\) is:
 
   \begin{align*}
     M &= 1.96 \times \dfrac{\sigma}{\sqrt{n}} \\
@@ -183,12 +184,57 @@
 
   \subsection*{General case}
 
-  A confidence interval of \(C\)% is given by
+  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}
 
-  A 95% confidence interval for \(\mu\) will have \(M\) when
-  \[ \overline{x} \pm k \dfrac{\sigma}{\sqrt{n}} \]
+  Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true.
 
-  where \(k\) is such that \(\Pr(-k < Z < k) = \frac{C}{100}\)
+  % 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}