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source/statistics/tests/parametric/ztest.rst
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Z-Test 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 z-test is used to assess if the mean :math:`\overline{x}` of sample :math:`X` differs from the one of a known population.
The *significance level* is determined by a *p-value* threshold chosen prior doing the test. The *significance level* of this difference is determined by a *p-value* threshold chosen prior doing the test.
Conditions for using a z-test: Conditions for using a z-test:
@ -16,8 +16,8 @@ Conditions for using a z-test:
To perform a z-test, you should compute the *standard score* (or *z-score*) of your sample. To perform a z-test, you should compute the *standard score* (or *z-score*) of your sample :math:`X`.
It characterizes how far from the population mean :math:`\mu` your sample mean :math:`\overline{x}` is, in unit of standard deviation :math:`\sigma`. 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: It is computed as follow:
.. math:: .. math::
@ -33,24 +33,32 @@ It is computed as follow:
However, if :math:`n` is sufficiently large, the distribution followed by :math:`Z` 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 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>`__). 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>`. 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)`:
That *p-value* is computed as follow:
* Left "tail" of the :math:`\mathcal{N}(0,1)` distribution: * Left "tail" of the :math:`\mathcal{N}(0,1)` distribution:
.. math:: .. math::
\alpha=P(\mathcal{N}(0,1)<Z\sigma)=P(\mathcal{N}(0,1)<Z\times 1)=P(\mathcal{N}(0,1)<Z) \alpha &= P(\mathcal{N}(0,1)<Z\sigma)
&=P(\mathcal{N}(0,1)<Z\times 1)
&=P(\mathcal{N}(0,1)<Z)=\Phi_{0,1}(Z)
* Right "tail" of the :math:`\mathcal{N}(0,1)` distribution: * Right "tail" of the :math:`\mathcal{N}(0,1)` distribution:
.. math:: .. 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) \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)=1-\Phi_{0,1}(Z)
.. image:: ../../figures/normal_law_tails.svg .. image:: ../../figures/normal_law_tails.svg
:align: center :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 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.
The following code shows you how to obtain the *p-value* in R: The following code shows you how to obtain the *p-value* in R:
@ -64,7 +72,7 @@ Output example:
Alpha approximated is 0.0359588035958804 Alpha approximated is 0.0359588035958804
Alpha from built-in CDF 0.0359303191129258 Alpha from built-in CDF 0.0359303191129258
If the :math:`\alpha` value given by the test is lower or equal to the *p-value* threshold chosen prior the test, If the :math:`\alpha` value given by the test is lower or equal to the *p-value* threshold chosen initially,
:math:`H_0` is rejected and :math:`H_1` is considered accepted. :math:`H_0` is rejected and :math:`H_1` is considered accepted.
An alternative way of doing the z-test is to build a **rejection region** from the *p-value*. An alternative way of doing the z-test is to build a **rejection region** from the *p-value*.