CH4/templates/02_stat.proporion_conf_int.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Доверительные интервалы для доли "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Генерация данных"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "np.random.seed(1)\n",
    "\n",
    "statistical_population = np.random.randint(2, size = 100000) \n",
    "\n",
    "random_sample = np.random.choice(statistical_population, size = 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.49770999999999999"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#истинное значение доли\n",
    "statistical_population.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Точечная оценка доли"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.502"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random_sample.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Доверительный интервал для доли"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from statsmodels.stats.proportion import proportion_confint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Доверительный интервал на основе нормального распределения"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\hat{p}\\pm z_{1-\\frac{\\alpha}{2}} \\sqrt{\\frac{\\hat{p}\\left(1-\\hat{p}\\right)}{n}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "normal_interval = proportion_confint(sum(random_sample), \n",
    "                                     len(random_sample), \n",
    "                                     method = 'normal')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "normal_interval [0.471010, 0.532990] with width 0.061979\n"
     ]
    }
   ],
   "source": [
    "print 'normal_interval [%f, %f] with width %f' % (normal_interval[0],\n",
    "                                                  normal_interval[1], \n",
    "                                                  normal_interval[1] - \n",
    "                                                  normal_interval[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Доверительный интервал Уилсона"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac1{ 1 + \\frac{z^2}{n} } \\left( \\hat{p} + \\frac{z^2}{2n} \\pm z \\sqrt{ \\frac{ \\hat{p}\\left(1-\\hat{p}\\right)}{n} + \\frac{\n",
    "z^2}{4n^2} } \\right), \\;\\; z \\equiv z_{1-\\frac{\\alpha}{2}}$$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "wilson_interval = proportion_confint(sum(random_sample), \n",
    "                                     len(random_sample), \n",
    "                                     method = 'wilson')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "wilson_interval [0.471062, 0.532922] with width 0.061860\n"
     ]
    }
   ],
   "source": [
    "print 'wilson_interval [%f, %f] with width %f' % (wilson_interval[0],\n",
    "                                                  wilson_interval[1],\n",
    "                                                  wilson_interval[1] - \n",
    "                                                  wilson_interval[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Размер выборки для интервала заданной ширины"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from statsmodels.stats.proportion import samplesize_confint_proportion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9602"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_samples = int(np.ceil(samplesize_confint_proportion(\n",
    "            random_sample.mean(),0.01)))\n",
    "n_samples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "np.random.seed(1)\n",
    "random_sample = np.random.choice(statistical_population, size = n_samples)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "normal_interval = proportion_confint(sum(random_sample), len(random_sample), \n",
    "                                     method = 'normal')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "normal_interval [0.481877, 0.501876] with width 0.019999\n"
     ]
    }
   ],
   "source": [
    "print 'normal_interval [%f, %f] with width %f' % (normal_interval[0],\n",
    "                                                  normal_interval[1],\n",
    "                                                  normal_interval[1] \n",
    "                                                  - normal_interval[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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