\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
+ \begin{align*}
+ p &= \Pr(\overline{X} \lessgtr \mu(H_1)) \\
+ &= 2 \cdot \Pr(\overline{X} <> \mu(H_1) | \mu = 8)
+ \end{align*}
- \subsection*{Distribution of sample mean}
+ Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true.
- If \(X \sim \operatorname{N}(\mu, \sigma)\), then distribution of sample mean \(\overline{X}\) is also normal with \(\overline{X} \sim \operatorname{N}(\mu, \frac{\sigma}{\sqrt{n}}\).
+ \vspace{0.5em}
+ \begin{tabularx}{23em}{|l|X|}
+ \hline
+ \rowcolor{cas}
+ \(\boldsymbol{p}\) & \textbf{Conclusion} \\
+ \hline
+ \(> 0.05\) & insufficient evidence against \(H_0\) \\
+ \(< 0.05\) (5\%) & good evidence against \(H_0\) \\
+ \(< 0.01\) (1\%) & strong evidence against \(H_0\) \\
+ \(< 0.001\) (0.1\%) & very strong evidence against \(H_0\) \\
+ \hline
+ \end{tabularx}
\subsection*{Statistical significance}
\end{description}
\end{cas}
+ \subsection*{One-tail and two-tail tests}
+
+ \subsubsection*{One tail}
+
+ \begin{itemize}
+ \item \(\mu\) has changed in one direction
+ \item State ``\(H_1: \mu \lessgtr \) known population mean''
+ \end{itemize}
+
+ \subsubsection*{Two tail}
+
+ \begin{itemize}
+ \item Direction of \(\Delta \mu\) is ambiguous
+ \item State ``\(H_1: \mu \ne\) known population mean''
+ \end{itemize}
+
+ For two tail tests:
+ \begin{align*}
+ p\text{-value} &= \Pr(|\overline{X} - \mu| \ge |\overline{x}_0 - \mu|) \\
+ &= \left( |Z| \ge \left|\dfrac{\overline{x}_0 - \mu}{\sigma \div \sqrt{n}} \right| \right)
+ \end{align*}
+
+ \subsection*{Modulus notation for two tail}
+
+ \(\Pr(|\overline{X} - \mu| \ge a) \implies\) ``the probability that the distance between \(\overline{\mu}\) and \(\mu\) is \(\ge a\)''
+
+ \subsection*{Inverse normal}
+
+ \begin{cas}
+ \verb;invNormCdf("L", ;\(\alpha\)\verb;, ;\(\dfrac{\sigma}{n^\alpha}\)\verb;, ;\(\mu\)\verb;);
+ \end{cas}
+
+ \subsection*{Errors}
+
+ \begin{description}[labelwidth=2.5cm, labelindent=0.5cm]
+ \item [Type I error] \(H_0\) is rejected when it is \textbf{true}
+ \item [Type II error] \(H_0\) is \textbf{not} rejected when it is \textbf{false}
+ \end{description}
+
+% \subsection*{Using c.i. to find \(p\)}
+% need more here
+
\end{document}