[spec] inverse p-value and distance integral

diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex
index 38165bb9df4d4dae2fdf7f53ed72f5f5ff11ca58..d66ece975983a2cf40ff524d6d65e934a4a16a08 100644 (file)
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -1267,7 +1267,10 @@
         \end{align*}
 
         \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
         \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 \(\boldsymbol{r}(t_1)\):}
         \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
 
         \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 dadaff7a8849fd2a6550aea4bc5c624306fe3b5f..bc64a9807a62a8c9c24a221623bb197cd7aca407 100644 (file)
--- a/spec/statistics.tex
+++ b/spec/statistics.tex
@@ -202,6 +202,10 @@
     \hline
   \end{tabularx}
 
     \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.
   \subsection*{Significance level \(\alpha\)}
 
   The condition for rejecting the null hypothesis.