From 379fea44e47ed37a49e84b67b65603f4d4454605 Mon Sep 17 00:00:00 2001
From: Loic Guegan <manzerbredes@mailbox.org>
Date: Wed, 18 Oct 2023 21:34:59 +0200
Subject: [PATCH] Minor changes

---
 source/statistics/tests/parametric/ztest.rst | 32 ++++++++++++--------
 1 file changed, 20 insertions(+), 12 deletions(-)

diff --git a/source/statistics/tests/parametric/ztest.rst b/source/statistics/tests/parametric/ztest.rst
index 75b2807..740a232 100644
--- a/source/statistics/tests/parametric/ztest.rst
+++ b/source/statistics/tests/parametric/ztest.rst
@@ -1,8 +1,8 @@
 Z-Test
 -------
 
-The z-test is used to assess if the mean :math:`\overline{x}` of sample :math:`X` significantly differ from the one of a known population.
-The *significance level* is determined by a *p-value* threshold chosen prior doing the test.
+The z-test is used to assess if the mean :math:`\overline{x}` of sample :math:`X` differs from the one of a known population.
+The *significance level* of this difference is determined by a *p-value* threshold chosen prior doing the test.
 
 Conditions for using a z-test:
 
@@ -16,8 +16,8 @@ Conditions for using a z-test:
 
 
 
-To perform a z-test, you should compute the *standard score* (or *z-score*) of your sample.
-It characterizes how far from the population mean :math:`\mu` your sample mean :math:`\overline{x}` is, in unit of standard deviation :math:`\sigma`.
+To perform a z-test, you should compute the *standard score* (or *z-score*) of your sample :math:`X`.
+The *z-score*, noted :math:`Z`, characterizes how far from the population mean :math:`\mu` your sample mean :math:`\overline{x}` is, in unit of standard deviation :math:`\sigma`.
 It is computed as follow:
 
 .. math::
@@ -33,24 +33,32 @@ It is computed as follow:
    However, if :math:`n` is sufficiently large, the distribution followed by :math:`Z` is very close to a normal one.
    So close that, using z-test in place of the student test to compute *p-values* leads to nominal differences (`source <https://stats.stackexchange.com/questions/625578/why-is-the-sample-standard-deviation-used-in-the-z-test>`__).
 
-From :math:`Z`, the z-test *p-value* can be derived using the :math:`\mathcal{N}(0,1)` :ref:`CDF <CDF>`.
-That *p-value* is computed as follow:
+From :math:`Z`, a *p-value* can be derived using the :math:`\mathcal{N}(0,1)` :ref:`CDF <CDF>` noted :math:`\Phi_{0,1}(x)`:
 
 * Left "tail" of the :math:`\mathcal{N}(0,1)` distribution:
 
   .. math::
-     \alpha=P(\mathcal{N}(0,1)<Z\sigma)=P(\mathcal{N}(0,1)<Z\times 1)=P(\mathcal{N}(0,1)<Z)
+     \alpha &= P(\mathcal{N}(0,1)<Z\sigma)
+
+     &=P(\mathcal{N}(0,1)<Z\times 1)
+     
+     &=P(\mathcal{N}(0,1)<Z)=\Phi_{0,1}(Z)
 
 * Right "tail" of the :math:`\mathcal{N}(0,1)` distribution:
 
   .. math::
-     \alpha=1-P(\mathcal{N}(0,1)<Z\sigma)=1-P(\mathcal{N}(0,1)<Z\times 1)=1-P(\mathcal{N}(0,1)<Z)
+     \alpha &= 1-P(\mathcal{N}(0,1)<Z\sigma)
+
+     &=1-P(\mathcal{N}(0,1)<Z\times 1)
+
+     &=1-P(\mathcal{N}(0,1)<Z)=1-\Phi_{0,1}(Z)
 
 .. image:: ../../figures/normal_law_tails.svg
    :align: center
-      
-If a z-test is done over one tail (left or right) it is called a **one-tailed** z-test.
-If a z-test is done over both tails (left and right) it is called a **two-tailed** z-test.
+          
+|       
+| If the test is done over one tail (left OR right) it is called a **one-tailed** z-test.
+| If the test is done over both tails (left AND right) it is called a **two-tailed** z-test.
 
 The following code shows you how to obtain the *p-value* in R:
 
@@ -64,7 +72,7 @@ Output example:
    Alpha approximated is 0.0359588035958804
    Alpha from built-in CDF 0.0359303191129258
 
-If the :math:`\alpha` value given by the test is lower or equal to the *p-value* threshold chosen prior the test,
+If the :math:`\alpha` value given by the test is lower or equal to the *p-value* threshold chosen initially,
 :math:`H_0` is rejected and :math:`H_1` is considered accepted.
 
 An alternative way of doing the z-test is to build a **rejection region** from the *p-value*.