From 1fdb98af8987959c1f49e36934015cbbf7828ad5 Mon Sep 17 00:00:00 2001
From: Loic Guegan <manzerbredes@mailbox.org>
Date: Wed, 18 Oct 2023 20:32:31 +0200
Subject: [PATCH] Minor changes

---
 source/statistics/code/ztest_pvalue.R           |  4 ----
 source/statistics/code/ztest_rejection_region.R |  7 +++++++
 source/statistics/tests/parametric/ztest.rst    | 17 +++++++++++++----
 3 files changed, 20 insertions(+), 8 deletions(-)
 create mode 100644 source/statistics/code/ztest_rejection_region.R

diff --git a/source/statistics/code/ztest_pvalue.R b/source/statistics/code/ztest_pvalue.R
index 4396556..9185fd0 100644
--- a/source/statistics/code/ztest_pvalue.R
+++ b/source/statistics/code/ztest_pvalue.R
@@ -1,10 +1,6 @@
-
-
 N=rnorm(9999999,mean=0,sd=1)
 cdf=ecdf(N)
 zscore=-1.8
 
 message(paste0("Alpha approximated is ",cdf(zscore)))
 message(paste0("Alpha from built-in CDF ",pnorm(zscore,mean=0,sd=1)))
-message(paste0("Bonus: zscore from p-value ",qnorm(0.02275,mean=0,sd=1)))
-
diff --git a/source/statistics/code/ztest_rejection_region.R b/source/statistics/code/ztest_rejection_region.R
new file mode 100644
index 0000000..309f110
--- /dev/null
+++ b/source/statistics/code/ztest_rejection_region.R
@@ -0,0 +1,7 @@
+alpha=0.0003
+lower=qnorm(alpha,mean=0,sd=1)
+upper=qnorm(1-alpha,mean=0,sd=1)
+
+message(paste0("Rejection region left tail Z in ]-inf,",lower,"]"))
+message(paste0("Rejection region right tail Z in [",lower,",+inf["))
+
diff --git a/source/statistics/tests/parametric/ztest.rst b/source/statistics/tests/parametric/ztest.rst
index 85bff69..8f71203 100644
--- a/source/statistics/tests/parametric/ztest.rst
+++ b/source/statistics/tests/parametric/ztest.rst
@@ -63,15 +63,24 @@ Output example:
                 
    Alpha approximated is 0.0359588035958804
    Alpha from built-in CDF 0.0359303191129258
-   Bonus: zscore from p-value -2.0000024438996
 
 If the :math:`\alpha` value given by the test is lower or equal to the *p-value* threshold chosen prior the test,
 :math:`H_0` is rejected and :math:`H_1` is considered accepted.
 
-Using the z-score from *p-value* function allows you to create a *rejection region*.
-It is an interval :math:`[\mathrm{zscore}_{min},\mathrm{zscore}_{max}]`.
-If the sample z-score is not part of the interval, then :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*.
+This is done by using the reverse :ref:`CDF <CDF>` function :math:`\Phi_{0,1}^{-1}(x)` as in the following code:
 
+.. literalinclude:: ../../code/ztest_rejection_region.R
+   :language: R
+
+Output:
+
+.. code-block:: console
+
+   Rejection region left tail Z in ]-inf,-3.43161440362327]
+   Rejection region right tail Z in [-3.43161440362327,+inf[
+
+Thus, if the z-score if part of one of the rejection regions, :math:`H_0` is rejected and :math:`H_1` is considered accepted.
 
 One-tailed vs Two-tailed
 ========================