{
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  {
   "cell_type": "markdown",
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    "collapsed": true
   },
   "source": [
    "## <center> Проверка статистической значимости желания британцев покинуть ЕС</center>\n",
    "<img src='img/brexit.jpg' width=60%>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**По данным, представленным [BBC](http://www.bbc.com/news/politics/eu_referendum/results), за выход Британии из ЕС проголосовали 17.410.742 чел. (51.9%), против - 16.141.241 (48.1%).**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "N_LEAVE, N_REMAIN = 17410742, 16141241"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "referendum_results = np.array([1] * N_LEAVE + [0] * N_REMAIN)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Точечная оценка доли желающих покинуть ЕС:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.51891841981441156"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(referendum_results)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Доверительный интервал на основе нормального распределения для доли желающих покинуть ЕС:**\n",
    "\n",
    "$$\\hat{p}\\pm z_{1-\\frac{\\alpha}{2}} \\sqrt{\\frac{\\hat{p}\\left(1-\\hat{p}\\right)}{n}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "normal_interval [0.518749, 0.519087] with width 0.000338\n"
     ]
    }
   ],
   "source": [
    "from statsmodels.stats.proportion import proportion_confint\n",
    "\n",
    "normal_interval = proportion_confint(np.sum(referendum_results), \n",
    "                                     referendum_results.shape[0], \n",
    "                                     method = 'normal')\n",
    "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": [
    "**У нас выборка большая, поэтому доверительный интервал Уилсона получается точно таким же.**\n",
    "\n",
    "$$\\frac1{ 1 + \\frac{z^2}{n} } \\left( \\hat{p} + \\frac{z^2}{2n} \\pm z \\sqrt{ \\frac{ \\hat{p}\\left(1-\\hat{p}\\right)}{n} + \\frac{\n",
    "z^2}{4n^2} } \\right), \\;\\; z \\equiv z_{1-\\frac{\\alpha}{2}}$$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "wilson_interval [0.518749, 0.519087] with width 0.000338\n"
     ]
    }
   ],
   "source": [
    "wilson_interval = proportion_confint(np.sum(referendum_results), \n",
    "                                     referendum_results.shape[0],\n",
    "                                     method = 'wilson')\n",
    "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": [
    "**Одновыборочный критерий Стьюдента говорит о том, что отличие выборочного среднего от 0.5 статистически значимо.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ttest_1sampResult(statistic=219.32343584383185, pvalue=0.0)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.stats import ttest_1samp\n",
    "\n",
    "ttest_1samp(referendum_results, 0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Это же подтверждают критерий знаков, критерий знаковых рангов Вилкоксона и биномиальный критерий для доли:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.stats import wilcoxon, binom_test\n",
    "from statsmodels.stats.descriptivestats import sign_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(634750.5, 1.8774494541967369e-322)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sign_test(referendum_results, 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "WilcoxonResult(statistic=270785329886072.0, pvalue=0.0)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wilcoxon(referendum_results - 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binom_test(N_LEAVE, N_LEAVE + N_REMAIN, 0.5, alternative = 'greater')"
   ]
  }
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