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Loïc Guégan 2023-10-19 10:30:58 +02:00
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35
source/statistics/tests/parametric/ztest.rst
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@ -14,24 +14,17 @@ 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.
To perform a z-test, you should compute the *standard score* (or *z-score*) of your sample :math:`X`.
To perform a z-test with a sample :math:`X` of size :math:`n`, you should compute the sample *standard score* (or *z-score*).
The *z-score*, noted :math:`Z`, 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::
Z=\frac{\overline{x}-\mu}{\sigma}
Z=\frac{\overline{x}-\mu}{\sigma_\overline{x}}=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}
.. 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}}}
In this case, :math:`Z` technically follow a t-distribution (student test).
However, if :math:`n` is sufficiently large, the distribution followed by :math:`Z` is very close to a normal one.
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>`__).
The SEM is used in the denominator to account for inaccuracies of :math:`\overline{x}`.
The more samples are collected, the more the denominator tends toward :math:`\sigma`.
See :ref:`SEM <SEM>` for more details.
From :math:`Z`, a *p-value* can be derived using the :math:`\mathcal{N}(0,1)` :ref:`CDF <CDF>` noted :math:`\Phi_{0,1}(x)`:
@ -100,7 +93,7 @@ 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)*.
This exercice is inpired from `this video <https://www.youtube.com/watch?v=bB-J6_wcGgE>`__.
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.
@ -115,9 +108,9 @@ Hypothesis:
* :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
Z=\frac{29.5-32}{\frac{9}{\sqrt{25}}}=-1.388889
From this z-score, the *p-value* is 0.3905915. As it is greater than 0.05, we cannot reject :math:`H_0`.
From this z-score, the *p-value* is 0.08243327. As it is greater than 0.05, we cannot reject :math:`H_0`.
Thus, the boys from the are not underfed.
Two-tailed
@ -127,7 +120,7 @@ This exercice is inpired from `this website <https://www.mathandstatistics.com/l
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.
with :math:`\alpha=0.05`. A sample of 100 students reveals that in average, they received $6873.
.. note::
@ -142,12 +135,12 @@ Hypothesis:
* :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`.
Z=\frac{6873-6800}{\frac{400}{\sqrt{100}}}=1.825
| The *p-value* associated with the left tail (using :math:`-Z` with the CDF) is 0.03400051.
| Thus, as we are doing a *two-tailed* z-test the *p-value* is :math:`2\times 0.03400051 = 0.06800103`.
| 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`.
Since :math:`0.06800103 >> 0.05` we cannot reject the null hypothesis :math:`H_0`.
Thus, Michigan receives the same amount of public school funding per student.