Minor changes

source/index.rst

@@ -18,10 +18,7 @@ Welcome to ScienceNotes's documentation!
    statistics/probability_distribution_functions.rst
 
    statistics/distributions/index.rst
-   statistics/tests_parametric/index.rst
+   statistics/tests/index.rst
-   statistics/tests_non_parametric/index.rst
-
-   
 
 
 Indices and tables

source/statistics/tests/index.rst

@@ -0,0 +1,32 @@
+Statistical Tests
+------------------
+
+
+Statistical tests are used to decide whether a hypotesis is true or not. To do so, two hypothesis must be formulated:
+
+#. :math:`H_0` : The one that is true at first glance (nothing particular, null, boring):
+   
+   - The mean of two populations are equals
+   - The average weight of the children in that school is not different from the one of the country
+
+#. :math:`H_1` : The one that says: "Something strange is happening"
+
+   - The mean of two populations are differents
+   - The average weight of the children in that school differ from the one of the country
+
+
+A statistical test usually works with a **p-value** noted :math:`\alpha`.
+It corresponds to the probability of obtaining the results that you have under the assumption that the null hypothesis is correct.
+In other words, how lucky you are of obtaining these results.
+     
+Prior performing a statistical test, you must choose a minimal *p-value*.
+:math:`H_0` will be rejected (meaning :math:`H_1` considered true) if the *p-value* obtained from the statistical test
+is lower or equal to the one you choose initially.
+The lower your initial *p-value* is, the more difficult it is to reject the null hypothesis.
+
+.. toctree::
+   :maxdepth: 2
+   :caption: Categories
+              
+   parametric/index
+   non-parametric/index

source/statistics/tests/parametric/index.rst

@@ -2,8 +2,8 @@ Parametric Tests
 -----------------
 
 .. toctree::
-   :maxdepth: 2
+   :maxdepth: 1
-   :caption: Statistics
 
-   ttest
    ztest
+   ttest
+

source/statistics/tests/parametric/ztest.rst

@@ -0,0 +1,48 @@
+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.
+
+Conditions for using a z-test:
+
+#. Population is normally distributed
+#. Population :math:`\mu` and :math:`\sigma` is known
+#. Sample size is greater than 30 (see note below)
+
+.. note::
+   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 is computed as follow:
+
+.. math::
+   Z=\frac{\overline{x}-\mu}{\sigma}
+
+.. note::
+   The following formula can also be seen, when the original population :math:`\sigma` is unknown:
+   
+   .. 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.
+   However, if :math:`n` is sufficiently large, the sample distribution 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>`__).
+   
+
+   
+One tailed two tailed:
+https://stats.oarc.ucla.edu/other/mult-pkg/faq/general/faq-what-are-the-differences-between-one-tailed-and-two-tailed-tests/
+
+example 2 tailed https://www.mathandstatistics.com/learn-stats/hypothesis-testing/two-tailed-z-test-hypothesis-test-by-hand
+
+
+
+Examples
+========

source/statistics/tests_parametric/ztest.rst

@@ -1,7 +0,0 @@
-Z-Test
--------
-
-One tailed two tailed:
-https://stats.oarc.ucla.edu/other/mult-pkg/faq/general/faq-what-are-the-differences-between-one-tailed-and-two-tailed-tests/
-
-example 2 tailed https://www.mathandstatistics.com/learn-stats/hypothesis-testing/two-tailed-z-test-hypothesis-test-by-hand