Intervalo-de-confianca.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Intervalos de Confiança"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Francisco A. Rodrigues, University of São Paulo.<br> \n",
    "https://sites.icmc.usp.br/francisco<br>\n",
    "francisco@icmc.usp.br"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Esse notebook é relacionado à aula: \n",
    "https://www.youtube.com/watch?v=AkmyfLc-EOs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos interpretaro intervalo de confiança de $(1-\\alpha)100\\%$ através de simulações. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1008x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Nível de confiança: 0.95\n",
      "Fraçao de intervalos que contém a média: 0.94\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "n = 50 # tamanho da amostra\n",
    "Ns = 100 # numero de intervalos\n",
    "mu = 2 # media populacional\n",
    "sigma = 2 # desvio padrão populacional\n",
    "beta = 0.95 # nivel de confianca\n",
    "zalpha = 1.96 # valor de z (a partir de beta)\n",
    "c = 0 # conta o numero de intervalos que contem a media\n",
    "plt.figure(figsize=(14,10))\n",
    "for s in range(1,Ns):\n",
    "    x = np.random.normal(mu, sigma, n) # sorteia uma amostra de tamanho n\n",
    "    IC1 = np.mean(x) - zalpha*sigma/np.sqrt(n) #intervalo inferior\n",
    "    IC2 = np.mean(x) + zalpha*sigma/np.sqrt(n) #intervalo superior\n",
    "    if(mu > IC1 and mu < IC2):\n",
    "        c =  c + 1\n",
    "        # mostra o intervalo em cinza se continar a media\n",
    "        plt.vlines(s, ymin=IC1, ymax=IC2, color = 'gray')\n",
    "        plt.plot(s,np.mean(x), 'o', color = 'gray', \n",
    "                 markersize=5)\n",
    "    else:\n",
    "        # mostra o intervalo que nao contem a media\n",
    "        plt.vlines(s, ymin=IC1, ymax=IC2, color = 'black', linestyles = 'dashed')\n",
    "        plt.plot(s,np.mean(x), 'o', color = 'black',\n",
    "                 markersize=5)\n",
    "plt.axhline(y = mu, color = 'black') # mostra a media populacional\n",
    "plt.xlabel('Amostra', fontsize=20)\n",
    "plt.show()\n",
    "print('Nível de confiança:', beta)\n",
    "print('Fraçao de intervalos que contém a média:', c/Ns)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculo do Intervalo de confiança"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos implementar uma função para calcular o intervalo de confiança automaticamente. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy.stats\n",
    "import numpy as np\n",
    "\n",
    "def confident_interval(Xs, n, confidence = 0.95, sigma = -1, s = -1):\n",
    "    zalpha = abs(scipy.stats.norm.ppf((1 - confidence)/2.))\n",
    "    if(sigma != -1): # se a variancia eh conhecida\n",
    "        IC1 = Xs - zalpha*sigma/np.sqrt(n)\n",
    "        IC2 = Xs + zalpha*sigma/np.sqrt(n)\n",
    "    else: # se a variancia eh desconhecida\n",
    "        if(n >= 50): # se o tamanho da amostra eh maior do que 50\n",
    "            # Usa a distribuicao normal\n",
    "            IC1 = Xs - zalpha*s/np.sqrt(n)\n",
    "            IC2 = Xs + zalpha*s/np.sqrt(n)\n",
    "        else: # se o tamanho da amostra eh menor do que 50\n",
    "            # Usa a distribuicao t de Student\n",
    "            talpha = scipy.stats.t.ppf((1 + confidence) / 2., n-1)\n",
    "            IC1 = Xs - talpha*s/np.sqrt(n)\n",
    "            IC2 = Xs + talpha*s/np.sqrt(n)\n",
    "    return [IC1, IC2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exemplo**: Em uma empresa de distribuição de alimentos pela internet, verificou-se que o tempo necessário para uma entrega tem distribuição normal com média $\\mu = 30$ minutos e desvio padrão $\\sigma = 10$ minutos. Em uma amostra de 50 entregadores, observou-se um tempo médio de entrega $\\bar{X}_{50} = 25$ minutos. Determine o intervalo de 95\\% de confiança para a média $\\mu$ de todos os entregadores da empresa."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Confidence interval: [22.228192351300645, 27.771807648699355]\n"
     ]
    }
   ],
   "source": [
    "Xs = 25\n",
    "n = 50\n",
    "confidence =0.95\n",
    "sigma = 10\n",
    "IC = confident_interval(Xs,n, confidence, sigma)\n",
    "print('Confidence interval:', IC)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exemplo** Em um provedor de videos na Internet, verificou-se que para uma amostra de 15 usuários, o tempo médio de exibição é igual a $\\bar{X}_{15} = 39,3$ minutos e o desvio padrão da amostra $S_{15} = 2,6$ minutos. Encontre um intervalo de 90\\% para a média populacional $\\mu$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Confidence interval: [38.117602363950525, 40.48239763604947]\n"
     ]
    }
   ],
   "source": [
    "Xs = 39.3\n",
    "s = 2.6\n",
    "n = 15\n",
    "confidence =0.9\n",
    "\n",
    "IC = confident_interval(Xs,n, confidence, -1, s)\n",
    "print('Confidence interval:', IC)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para um conjunto de dados, temos a função abaixo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy.stats\n",
    "import numpy as np\n",
    "\n",
    "def confident_interval_data(X, confidence = 0.95, sigma = -1):\n",
    "    def S(X): #funcao para calcular o desvio padrao amostral\n",
    "        s = 0\n",
    "        for i in range(0,len(X)):\n",
    "            s = s + (X[i] - np.mean(X))**2\n",
    "        s = np.sqrt(s/(len(X)-1))\n",
    "        return s\n",
    "    n = len(X) # numero de elementos na amostra\n",
    "    Xs = np.mean(X) # media amostral\n",
    "    s = S(X) # desvio padrao amostral\n",
    "    zalpha = abs(scipy.stats.norm.ppf((1 - confidence)/2))\n",
    "    if(sigma != -1): # se a variancia eh conhecida\n",
    "        IC1 = Xs - zalpha*sigma/np.sqrt(n)\n",
    "        IC2 = Xs + zalpha*sigma/np.sqrt(n)\n",
    "    else: # se a variancia eh desconhecida\n",
    "        if(n >= 50): # se o tamanho da amostra eh maior do que 50\n",
    "            # Usa a distribuicao normal\n",
    "            IC1 = Xs - zalpha*s/np.sqrt(n)\n",
    "            IC2 = Xs + zalpha*s/np.sqrt(n)\n",
    "        else: # se o tamanho da amostra eh menor do que 50\n",
    "            # Usa a distribuicao t de Student\n",
    "            talpha = scipy.stats.t.ppf((1 + confidence) / 2., n-1)\n",
    "            IC1 = Xs - talpha*s/np.sqrt(n)\n",
    "            IC2 = Xs + talpha*s/np.sqrt(n)\n",
    "    return [IC1, IC2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Executando para um exemplo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Confidence interval: [1.0367568385224393, 4.9632431614775605]\n"
     ]
    }
   ],
   "source": [
    "X = [1, 2, 3, 4, 5]\n",
    "confidence = 0.95\n",
    "\n",
    "IC = confident_interval_data(X, confidence)\n",
    "print('Confidence interval:', IC)"
   ]
  }
 ],
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