[spec] p-values table and one/two-tail tests

diff --git a/spec/statistics.pdf b/spec/statistics.pdf
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+++ b/spec/statistics.tex
@@ -256,13 +256,26 @@
 
   \subsection*{\(p\)-value}
 
 
   \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}
 
 
   \subsection*{Statistical significance}
 


@@ -287,4 +300,46 @@
   \end{description}
   \end{cas}
 
   \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}
 \end{document}