From f2e4e86c4f621969ab5aa5c5c5f779ae07e06b89 Mon Sep 17 00:00:00 2001 From: Loic Guegan Date: Thu, 19 Oct 2023 10:01:43 +0200 Subject: [PATCH] Minor changes --- source/statistics/bayes_theorem.rst | 3 +- source/statistics/tests/parametric/ztest.rst | 68 +++++++++++++++++--- 2 files changed, 59 insertions(+), 12 deletions(-) diff --git a/source/statistics/bayes_theorem.rst b/source/statistics/bayes_theorem.rst index 8d310f4..ac652b2 100644 --- a/source/statistics/bayes_theorem.rst +++ b/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: diff --git a/source/statistics/tests/parametric/ztest.rst b/source/statistics/tests/parametric/ztest.rst index 740a232..a411df1 100644 --- a/source/statistics/tests/parametric/ztest.rst +++ b/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 `__ *(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 `__. + +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. +