Minor changes

source/statistics/bayes_theorem.rst

@@ -51,8 +51,7 @@ Anna says she is itchy. There is a test for Allergy to Cats, but this test is no
 
 If 1% of the population have the allergy, and Anna's test says *Yes*, what are the chances that she really has the allergy?
 
-Solution
+**--- Solution ---**
-^^^^^^^^^
 
 Let's draw the little diagram:
 

source/statistics/tests/parametric/ztest.rst

@@ -60,6 +60,9 @@ From :math:`Z`, a *p-value* can be derived using the :math:`\mathcal{N}(0,1)` :r
 | 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.
 
+| A *one-tailed* z-test checks whether :math:`\overline{x} < \mu` or :math:`\overline{x} > \mu`.
+| A *two-tailed* z-test checks whether :math:`\overline{x} \ne \mu`.
+
 The following code shows you how to obtain the *p-value* in R:
 
 .. literalinclude:: ../../code/ztest_pvalue.R
@@ -90,16 +93,61 @@ Output:
    
 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
-========================
-
    
-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
 ========
+
+One-tailed
+^^^^^^^^^^^
+
+This exercice is inpired from `this video <https://www.youtube.com/results?search_query=ztest>`__ *(be careful the video uses a wrong formula)*.
+
+A complain was registered stating that the boys in the municipal school are underfed.
+The average weight of boys of age 10 is 32kg with a standard deviation of 9kg.
+A sample of 25 boys of age 10 from the school is selected. Their average weight is 29.5kg.
+We want to check whether the complain is true or not with a confidence level of :math:`\alpha=0.05`.
+
+**--- Solution ---**
+
+Hypothesis:
+
+* :math:`H_0` : No significant difference (:math:`\overline{x} \ge 32`), the boys from the are not underfed
+* :math:`H_1` : There is significant difference (:math:`\overline{x} < 32`), the boys from the are underfed
+
+.. math::
+   Z=\frac{29.5-32}{9}=-0.2777778
+
+From this z-score, the *p-value* is 0.3905915. As it is greater than 0.05, we cannot reject :math:`H_0`.
+Thus, the boys from the are not underfed.
+  
+Two-tailed
+^^^^^^^^^^^
+
+This exercice is inpired from `this website <https://www.mathandstatistics.com/learn-stats/hypothesis-testing/two-tailed-z-test-hypothesis-test-by-hand>`__.
+
+The USA mean public school yearly funding is $6800 per student per year, with a standard deviation of $400.
+We want to assess if a certain state in the USA, Michigan, receives a significantly different amount of public school funding (per student) than the USA average,
+with :math:`\alpha=0.05`. A sample of 1000 students reveals that in average, they received $6873.
+
+.. note::
+   
+   Notice, we are not saying **"significantly lower amount"** or **"significantly higher amount"** but **"significantly different amount"**.
+   This is a sign that a two-tailed z-test is required since we should check for both, lower and higher.
+
+**--- Solution ---**
+
+Hypothesis:
+
+* :math:`H_0` : No significant difference (:math:`\overline{x} = 6800`), Michigan receives the same amount of public school funding per student
+* :math:`H_1` : There is significant difference (:math:`\overline{x} \ne 6800`), Michigan do not receives the same amount of public school funding per student
+
+.. math::
+   Z=\frac{6873-6800}{400}=0.1825
+
+| The *p-value* associated with the left tail (using :math:`-Z` with the CDF) is 0.4275952.
+| Thus, as we are doing a *two-tailed* z-test the *p-value* is :math:`2\times 0.4275952 = 0.8551904`.
+| We multiply by two has the two tails of the normal law are symetric.
+
+Since :math:`0.8551904 >> 0.05` we cannot reject the null hypothesis :math:`H_0`.
+Thus, Michigan receives the same amount of public school funding per student.
+