From a66bb661b80b9b6b90e0865b02f753013bb581d9 Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Fri, 11 Oct 2019 10:20:08 +1100
Subject: [PATCH] [spec] inverse p-value and distance integral

---
 spec/spec-collated.tex | 5 ++++-
 spec/statistics.tex    | 4 ++++
 2 files changed, 8 insertions(+), 1 deletion(-)

diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex
index 38165bb..d66ece9 100644
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -1267,7 +1267,10 @@
         \end{align*}
 
         \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
-        \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
+        \begin{align*}
+          &= \int^{b}_{a}{\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}} \> dt \tag{2D} \\
+          &= \int^{t=b}_{t=a}{\dfrac{dx}{dt}} \> dt \tag{linear}
+        \end{align*}
 
         \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
         \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
diff --git a/spec/statistics.tex b/spec/statistics.tex
index dadaff7..bc64a98 100644
--- a/spec/statistics.tex
+++ b/spec/statistics.tex
@@ -202,6 +202,10 @@
     \hline
   \end{tabularx}
 
+  \subsubsection*{Finding \(n\) for a given \(p\)-value}
+
+  Find \(c\) such that \(\Pr(Z \lessgtr c)\) such that \(c = \alpha\) (use \texttt{invNormCdf} on CAS).
+
   \subsection*{Significance level \(\alpha\)}
 
   The condition for rejecting the null hypothesis.
-- 
2.43.2