Minor changes

source/statistics/tests/parametric/ztest.rst

@@ -2,7 +2,7 @@ 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.
+The *significance level* is determined by a *p-value* threshold chosen prior doing the test.
 
 Conditions for using a z-test:
 
@@ -14,12 +14,10 @@ Conditions for using a z-test:
    According to central limit theorem, a distribution is well approximated when reaching 30 samples.
    See `here <https://statisticsbyjim.com/basics/central-limit-theorem/>`__ for more infos.
 
-One-tailed vs Two-tailed
-========================
 
 
 To perform a z-test, you should compute the *standard score* (or *z-score*) of your sample.
-It corresponds to the projection of the sample mean :math:`\overline{x}` under the original population distribution.
+It 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::
@@ -31,10 +29,34 @@ It is computed as follow:
    .. math::
          Z=\frac{\overline{x}-\mu}{\mathrm{SEM}}=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}
 
-   This formula originate from the t-test and :math:`Z` technically follow a t-distribution.
+   In this case, :math:`Z` technically follow a t-distribution (student test).
-   However, if :math:`n` is sufficiently large, the sample distribution is very close to a normal one.
+   However, if :math:`n` is sufficiently large, the distribution followed by :math:`Z` is very close to a normal one.
-   So close that, using the normal in place of the student-t 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>`__).
+   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:
+
+* 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)
+
+* 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)
+
+.. 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 :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.
+
+One-tailed vs Two-tailed
+========================
 
    
 One tailed two tailed:

test.R

@@ -0,0 +1,8 @@
+
+
+cdf=ecdf(N)
+initial_alpha=0.05
+
+print(round(pnorm(1.01,mean=0,sd=1),digits=4))
+
+