.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statistics/confidence_intervals/plot_01_coverage_simulation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_statistics_confidence_intervals_plot_01_coverage_simulation.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_statistics_confidence_intervals_plot_01_coverage_simulation.py:


Confirming a 95% confidence interval's coverage by simulation
====================================================================

Builds many 95% confidence intervals from repeated samples of a known
population, and checks that close to 95% of them actually contain the
true parameter -- the frequentist interpretation of "95% confidence."

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.statistics import mean_confidence_interval, proportion_confidence_interval








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Mean confidence interval coverage
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-33

.. code-block:: Python


    rng = np.random.default_rng(0)
    true_mean = 50.0
    n_trials = 2000
    intervals = []
    for _ in range(n_trials):
        sample = rng.normal(loc=true_mean, scale=10.0, size=40)
        result = mean_confidence_interval(sample, sigma=10.0)
        intervals.append((result.lower, result.upper))
    intervals = np.array(intervals)
    hits = (intervals[:, 0] < true_mean) & (true_mean < intervals[:, 1])

    print(f"mean CI empirical coverage: {hits.mean():.4f} (target 0.95)")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    mean CI empirical coverage: 0.9445 (target 0.95)




.. GENERATED FROM PYTHON SOURCE LINES 34-39

The first 100 intervals
-----------------------------------------------------

Each interval either contains the true mean or it does not; "95%" is
the long-run hit rate of the procedure, visible as the few misses.

.. GENERATED FROM PYTHON SOURCE LINES 39-49

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6, 7))
    for i, ((lo, hi), hit) in enumerate(zip(intervals[:100], hits[:100])):
        ax.plot([lo, hi], [i, i], color="C0" if hit else "C3", lw=1.5)
    ax.axvline(true_mean, color="k", ls="--", label="true mean")
    ax.set_xlabel("95% confidence interval for the mean")
    ax.set_ylabel("repeated sample")
    ax.set_title(f"{np.sum(~hits[:100])} of 100 intervals miss (red)")
    ax.legend()




.. image-sg:: /api/gallery/statistics/confidence_intervals/images/sphx_glr_plot_01_coverage_simulation_001.png
   :alt: 5 of 100 intervals miss (red)
   :srcset: /api/gallery/statistics/confidence_intervals/images/sphx_glr_plot_01_coverage_simulation_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11801be00>



.. GENERATED FROM PYTHON SOURCE LINES 50-52

Proportion confidence interval
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 52-55

.. code-block:: Python


    result = proportion_confidence_interval(successes=42, n=100)
    print(f"proportion estimate: {result.estimate:.3f}, 95% CI: ({result.lower:.3f}, {result.upper:.3f})")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    proportion estimate: 0.420, 95% CI: (0.323, 0.517)





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.082 seconds)


.. _sphx_glr_download_api_gallery_statistics_confidence_intervals_plot_01_coverage_simulation.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_coverage_simulation.ipynb <plot_01_coverage_simulation.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_coverage_simulation.py <plot_01_coverage_simulation.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_coverage_simulation.zip <plot_01_coverage_simulation.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_