T-Test_normality_Precision.ipynb

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    "# Hypothesis testing\n",
    "\n",
    "### Example 1\n",
    "\n",
    "US male adult population height. \n",
    "\n",
    "$H_0: \\mu = 72$\n",
    "\n",
    "and your alternative hypothesis is\n",
    "\n",
    "$H_a: \\mu < 72$ \n",
    "    \n",
    "$1-\\alpha = 95\\%$\n",
    "\n"
   ]
  },
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "mu_real = 71\n",
    "s_real = 2.8\n",
    "n_samples = 15\n",
    "values= np.random.normal(mu_real,s_real,n_samples)\n",
    "n, bins, patches = plt.hist(values,7, alpha=0.75)\n",
    "plt.xlabel('height in')\n",
    "plt.ylabel('count')\n",
    "plt.show()"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "## Testing a mean of a population\n",
    "\n",
    "You know the population is distributed normal, but you don't know the variance? \n",
    "\n",
    "In this case is the __student's t test__. The __t-distribution__ has parameter, $\\nu$:\n",
    "\n",
    "$\\nu = n - 1$ \n",
    "\n",
    "Our test statistics is based on the mean of the sample, $\\hat X$ computed as:\n",
    "\n",
    "$\\hat X = \\frac{1}{n} \\sum_{i=1}^{n} X_i$ \n",
    "\n",
    "Sample variance:\n",
    "\n",
    "$S^2 = \\frac{1}{n-1} \\sum_{i=1}^{n} (X_i-\\mu)^2$\n",
    "\n",
    "So then, the __t-score or t-test statistics t__:\n",
    "\n",
    "$\\hat t = \\frac{\\hat X - \\mu}{\\frac{S}{\\sqrt n}}$\n",
    "\n",
    "And then:\n",
    "\n",
    "$t \\sim t(n-1)$"
   ]
  },
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   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
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    "deletable": true,
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample mean:  71.911\n",
      "Sample standard deviation:  2.753\n",
      "t-statistics:  -0.125\n"
     ]
    }
   ],
   "source": [
    "x_bar = np.mean(values)\n",
    "S = np.sqrt(np.var(values,ddof=1)) #ddof is delta degrees of freedom. It is the numerator n-ddof, which in our case is 1\n",
    "t_stat = np.sqrt(n_samples)*(x_bar-72)/S\n",
    "\n",
    "print(\"Sample mean: \",np.round(x_bar,3))\n",
    "print(\"Sample standard deviation: \",np.round(S,3) )\n",
    "print(\"t-statistics: \",np.round(t_stat,3) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    "editable": true
   },
   "source": [
    "$\\hat t = \\frac{\\hat X - \\mu}{\\frac{S}{\\sqrt n}} = \\frac{71.911 - 72}{\\frac{2.753}{\\sqrt{15}}} = -0.125$"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
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   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import t\n",
    "df = n_samples-1\n",
    "\n",
    "x = np.linspace(-4,4, 200)\n",
    "w1 = t.ppf(0.05,df)\n",
    "x1 = np.linspace(-4,w1, 200)\n",
    "w2 = t.ppf(0.95,df)\n",
    "x2 = np.linspace(w2,3, 200)\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "\n",
    "y = t.pdf(x,df)\n",
    "y1 = t.pdf(x1,df)\n",
    "\n",
    "ax.plot(x, y, 'b-')\n",
    "plt.fill_between(x1,y1, color = 'b',alpha=0.4)\n",
    "\n",
    "plt.xlabel('t')\n",
    "plt.ylabel('probability density function')\n",
    "\n",
    "plt.text(-1.1, 0.15, r'$1-\\alpha = 95\\%$', fontsize=18)\n",
    "plt.text(-3.2, 0.1, r'$\\alpha = 5\\%$', fontsize=18)\n",
    "#plt.text(2.1, 0.1, r'$\\alpha/2 = 5\\%$', fontsize=18)\n",
    "plt.text(-3, 0.02, r'Rejection', fontsize=14)\n",
    "t_crit =  np.round(t.ppf(0.05,df),3)\n",
    "plt.text(t_crit-0.05, -0.05, r'$t_{cr}$', fontsize=20)\n",
    "plt.text(t_stat, -0.015, r'$\\hat t$', fontsize=20)\n",
    "plt.text(0.5, 0.35, r'$H_0$', fontsize=20)\n",
    "\n",
    "plt.axvline(x=t_stat, color='r')\n",
    "plt.axvline(x=t_crit, color='b')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The critical t-statistics is:  -1.761\n"
     ]
    }
   ],
   "source": [
    "t_crit =  np.round(t.ppf(0.05,df),3)\n",
    "print(\"The critical t-statistics is: \", t_crit)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "How to define the p-value? It's the probability of a type I error given your data. In this case:\n",
    "\n",
    "$ p = P(t< \\hat t) = P(t\\leq -0.125 ) = 0.451$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p:  45.1 %\n"
     ]
    }
   ],
   "source": [
    "print(\"p: \",np.round(t.cdf(t_stat,df)*100,2),\"%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "You cannot reject the null hypothesis: two options\n",
    "* You lower your level of confidence, increasing $\\alpha$ to at least your p-value\n",
    "* You do more experiments"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_samples = 25\n",
    "values= np.random.normal(mu_real,s_real,n_samples)\n",
    "\n",
    "n_samples = n_samples\n",
    "n, bins, patches = plt.hist(values,7, alpha=0.75)\n",
    "plt.xlabel('height in')\n",
    "plt.ylabel('count')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample mean:  70.598\n",
      "Sample standard deviation:  2.749\n",
      "t-statistics:  -2.549\n"
     ]
    }
   ],
   "source": [
    "x_bar = np.mean(values)\n",
    "S = np.sqrt(np.var(values,ddof=1)) #ddof is delta degrees of freedom. It is the numerator n-ddof, which in our case is 1\n",
    "t_stat = np.sqrt(n_samples)*(x_bar-72)/S\n",
    "\n",
    "print(\"Sample mean: \",np.round(x_bar,3))\n",
    "print(\"Sample standard deviation: \",np.round(S,3) )\n",
    "print(\"t-statistics: \",np.round(t_stat,3) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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cdJ2tpKqeF661rHwgOho2bzaD8fKScuXgzjth6lQzfXleqD6z8i4nJYtfROR6r0diWT7y2Wdmor577vF1JJnXrx8cPgwLFvg6Eiu/c5JZNAPWishWEfldRDaIyO/eDsyyckJsrFngqFMnKFbM19FkXrt2UKqUaXOxLG9yUg3VzutRWJaPzJsHx4+b+v+8KDDQlIimTDFjLnyxTrh1dciwZKGqe4HiQHvXVty1z7LyvM8+g9BQaNPG15FkXa9ecOECfPutryOx8rMMMwsRGQx8CZR1bV+IyOPeDsyyvO34cfjvf82XbQGvzpLmXU2bmulJvvzS15FY+ZmTNosHgUaqOtI1CWBjIJdOs2ZZzs2eDZcvm8wiLxMxy64uWgQOl9C2rExzklkIZprwRPGufZaVp02fDpGRUK+eryO5cr16QUICfP21ryOx8isnmcVkYLWIjBaR0cAq4BOvRmVZXnboECxdCj16XNmaELlFzZom07O9oixvcdLA/TbwAHACs0DRA6rqeTV0y8ojZs40v8S7d/d1JNmnVy8zwHDrVl9HYuVHaWYWIlLU9VgS2AN8AXwO7HXty5CItHWNz9ghIsM8HB/oGrexTkSWi0hNt2PDXedtFZHbM/m+LCtdX38NtWtDrVq+jiT7JJaSbFWU5Q3plSwSV/ldC0S7bYmv0+VaQ3s8ZpxGTeBe98wg8R6qer2q1gXeAN52nVsTs2Z3LaAt8EHimtyWdaX274fly/NXqQKgYkUzv9WMGb6OxMqP0swsVPUu12OEqlZ12yJU1cl0aw2BHaq6S1VjgemYCQnd73Ha7WVhQF3POwLTVfWSqu4GdriuZ1lXLPHLNL9lFmAG6G3aBH/84etIrPzGyTiLxU72eVAR2O/2+oBrX8prPSoiOzEli0GZPLe/iESLSPTRo0cdhGRZppqmfn3TEyq/6dLFVEXZRZGs7JZem0Wwq22itIiUEJGSri0cqODg2p76mGiqHarjVfUaYCjw70yeO1FVo1Q1qkyZMg5Csq52O3fCmjWmfj8/Kl8ebrnFVkVZ2S+9ksUATPvEta7HxO17TFtERg4Ald1eVwIOppN+OtApi+daliOJX6J5cYZZp7p1M9VQmzb5OhIrP0mvzeJdVY0Ahri1VUSo6g2q+r6Da68BIkUkQkQCMQ3Wc9wTiIh7RcCdwHbX8zlADxEJEpEIIBL4NRPvy7I8mj7dLElapYqvI/GexKooW7qwspOTQXkJIlI88YWrSuqRjE5S1TjgMWABsBmYoaqbRORFEengSvaYiGwSkXXAU0Bf17mbgBnAH8APwKOqGp/qJpaVCZs3w++/598qqETlykHz5qbdQlNV3lpW1jiZPu0hVU2qdlLVv0XkIeCDjE5U1XnAvBT7Rro9H5zOuS8DLzuIz7Ic+fpr84u7a1dfR+J999wDjzxiqqJq1/Z1NFZ+4KRk4Sfyz4QIrvEOgd4LybK8Y8YM0/hbwUn3jDzu7rvBz89WRVnZx0lmsQCYISK3icitwDRM1ZBl5RmbN5vtaihVgFmjo3lzk1nYqigrOzjJLIYC/wMeBh4FFgPPejMoy8pus2ebx86dfRtHTrrnHjNP1IYNvo7Eyg+cTCSYoKoTVLWrqnZR1Y9sY7OV18yaBY0bmykxrhaJVVF2gJ6VHZyM4L5JRBaKyDYR2SUiu0VkV04EZ1nZYfduiIkxXUqvJmXLQsuWpmHfVkVZV8pJNdQnmAn+mgE3AlGuR8vKExKroO6+27dx+EK3brB9O2zc6OtIrLzOSWZxSlXnq+oRVT2euHk9MsvKJrNnQ926UNXJ9Jf5TKdOprvwzJm+jsTK65xkFktE5E0RaSIi9RM3r0dmWdng4EH45ZerrwoqUWgo3HyzabOxrCvhZFBeI9djlNs+BW7N/nAsK3t9+615vBqroBJ17QqDBsGWLXDttb6OxsqrnPSGaulhsxmFlSfMnm2+IGumXHbrKpKYUdrShXUlMixZiMhIT/tV9cXsD8eyss+xY/DTTzB0qK8j8a2KFU234VmzYMQIX0dj5VVO2izOuW3xmGVSw70Yk2Vli++/h/j4q7e9wl3Xrqb78C7b6d3KIifVUG+5bS8DLfCwap1l5TazZ0N4ONSr5+tIfM9WRVlXyknJIqVCwFXYCdHKS06dgoULzZekeFp38SoTEWGWkrWZhZVVTkZwbxCR313bJmAr8K73Q7OsrPvvf+HyZVsF5a5rV1i9Gg4c8HUkVl6U3hrcEa6ndwHtXVsboILDlfIsy2dmzzbrUTdu7OtIco/EjDNxRLtlZUZ6JYvEMZ+fqupe1/anawU8R0SkrYhsFZEdIjLMw/GnROQPV6llsYhUcTsWLyLrXNuclOdaVlrOnYP5880Ms35ZqWjNp6pXNwsh2dHcVlak13XWT0RGAdVF5KmUB1X17fQu7FokaTzQGjgArBGROar6h1uyGCBKVc+LyMPAG0B317ELqlo3E+/FsgBYsAAuXLBVUJ507QovvACHDpnlVy3LqfR+d/UALmIylCIetow0BHao6i5VjQWmAx3dE6jqElU973q5CqiUufAtK7VZs6BUKbMqnpVcly5mBtrvvvN1JFZek2bJQlW3Aq+LyO+qOj8L164I7Hd7fYB/pg7x5EHA/T7BIhINxAGvqWqqj7eI9Af6A4SFhWUhRCu/uXTJNG537QoFnExmc5WpVctUR82cCQMH+joaKy9xMs4iKxkFgKcOix5n1ReR+zBzT73ptjtMVaOAnsA4EbnGQ2wTVTVKVaPKlCmTxTCt/GTxYjh92lZBpUXEZKRLl8JxO3e0lQnebP47AFR2e10JOJgykYi0AkYAHVT1UuJ+VT3oetwFLAXs0CorQ7NmQdGicNttvo4k9+rSxYxs//57X0di5SXezCzWAJEiEiEigZg2kGS9mkSkHvARJqM44ra/hIgEuZ6XBm4C3BvGLSuVuDjzBXjXXRAU5Otocq969czIdtsrysoMJ4PyokXkUREpkZkLu7rYPgYsADYDM1R1k4i8KCIdXMneBEKAb1J0kb0OiBaR9cASTJuFzSysdC1bZqpWbBVU+hKrohYtgpMnfR2NlVc4aQLsATyA6foaDUwGflTNeFVfVZ0HzEuxb6Tb81ZpnPcLcL2D2CwryaxZULAgtG3r60hyvy5dYOxY0xngvvt8HY2VFzhp4N6hqiOA6sBXwKfAPhF5QURKejtAy3IiIcEsdNSuHRQq5Otocr+GDc3U5bYqynLKUZuFiNQB3sJUG80CugKngf95LzTLcm7VKvjrL1sF5ZSfn/lbLVgAZ8/6OhorL3DSZrEWeAfTYF1HVQep6mpVfQuws+NbucKsWRAYaBq3LWe6dIGLF2HevIzTWpaTkkU3Vb1NVb9K7NqaOMmgql7FKxtbuYWqmRyvVSvTbdZy5qaboGxZWxVlOeMks/D0UbIfLyvX+O032LPH9PCxnPP3N+t9zJtn5tKyrPSkN0X5tSLSBSgmIne7bfcDwTkWoWVlYOZMM7VHx44Zp7WS69LFzNK7YIGvI7Fyu/S6ztbArGVRHLOWRaIzwEPeDMqynFI1mUXLllDS9s3LtObNzaSLM2dCp06+jsbKzdKbSPB74HsRaaKqK3MwJstybMMG2LEDnnnG15HkTQEBpkQ2c6aZhNGOfLfSkl411LOupz1F5D8ptxyKz7LSNXOm6QZqfxVnXZcuZvLFxYt9HYmVm6VXDbXZ9RidE4FYVlbMnGmqUsqW9XUkeddtt0GxYuZveccdvo7Gyq3Sq4b6P9fj1MR9IuIHhKjq6RyIzbLS9ccfsHkzPPaYryPJ24KCoH17Mwnj5cumasqyUnIyKO8rESkqIoUxM79uFRFbQ2z53MyZZlK8zp19HUne16ULnDgBP/3k60is3MrJOIuarpJEJ8ykgGFAb69GZVkOzJxpBpaVL+/rSPK+22+HwoXtAD0rbU4yiwARCcBkFt+r6mXSWPHOsnLKtm2mJ5QdiJc9ChaEO+80kzHGx/s6Gis3cpJZfATsAQoDy0SkCmYSQcvymVmzzOPddsKZbNOlCxw5AitW+DoSKzdyMkX5f1S1oqreocZeoGUOxGZZaZo5Exo3hsqVM05rOXPHHRAcbKuiLM+cNHAHiUhPEXlOREaKyEjgOScXF5G2IrJVRHaIyDAPx58SkT9E5HcRWewqtSQe6ysi211b30y9Kytf27XLzAdlpyPPXiEhZuGo2bPN+iCW5c5JNdT3QEcgDjjntqVLRPyB8UA7oCZwr4jUTJEsBohS1TqYyQnfcJ1bEhgFNAIaAqMyu6yrlX8lVkHZzCL7dekCf/4Jv/7q60is3MbJsqqVVDUrC1U2BHao6i4AEZmOyXSS1tJW1SVu6VcBiQs83g4sVNUTrnMXAm2BaVmIw8pnZs2CBg0gIsLXkeQ/7dubcRaJ1XyWlchJyeIXEcnKetgVgf1urw+49qXlQWB+Zs4Vkf4iEi0i0UePHs1CiFZes28frF5te0F5S7Fi0Lq1yZDV9nm03DjJLJoBa11tD7+LyAYR+d3BeeJhn8ePn4jcB0Rhlm11fK6qTlTVKFWNKlOmjIOQrLxu9mzzaKugvKdLF7M+SEyMryOxchMnmUU7IBJog5mq/C6ST1melgOAe1+VSsDBlIlEpBUwAuiQuBKf03Otq8/MmVCnDkRG+jqS/KtjR7Mwku0VZblz0nV2L+aL+1bX8/NOzsOs2R0pIhEiEgj0AOa4JxCRephxHB1U9YjboQVAGxEp4WrYbuPaZ13F/vwTfvnFVkF5W6lSZn0QWxVluXPSdXYUMBQY7toVAHyR0XmqGgc8hvmS3wzMUNVNIvKiiHRwJXsTCAG+EZF1IjLHde4JYAwmw1kDvJjY2G1dvb7+2nx59ejh60jyvy5dzCj5TZt8HYmVWzjpDdUZqAf8BqCqB0WkiJOLq+o8zHxS7vtGuj1vlc65nwKfOrmPdXWYPt30grJVUN7XuTM88oipiqpd29fRWLmBk+qkWFVVXA3MrtlnLStH7dgBa9bYUkVOCQ2Fm2/+Z0yLZTnJLGaIyEdAcRF5CFgEfOzdsCwrua+/No/du/s2jqtJly6wcSNs3errSKzcwEkD91jM6OpZQA1gpKq+5+3ALMvdtGnQrJmdCyonde1q1guZPt3XkVi5gZOSBaq6UFWfUdUhqrrQ20FZlruNG01D6733+jqSq0uFCqZX1Jdf2l5RVjqZhYicEZHTaW05GaR1dZs2zfT7t11mc17PnrB9O0RH+zoSy9fSzCxUtYiqFgXGAcMw021UwnSjfSlnwrOudqqmGuS226BsWV9Hc/Xp0gUCA+Grr3wdieVrTqqhblfVD1T1jKqeVtUJgJ1swcoRa9aYKcltFZRvFC9uVtCbPt2uoHe1c5JZxItILxHxFxE/EekF2I+NlSOmTTO/bDt1yjjtq6++Srdu3ahatSoiQnh4uNfjy60OHz7MwIEDqVy5MoGBgYSFhTF48GBOnjyZKq2IeNxCQkIA6NULDh2CJUvg7NmzDBgwgNDQUEJDQ3n44Yc5dy71igXffvsthQsXZs+ePd5+q1YOcTIoryfwrmtTYIVrn2V5VXy86TJ7xx3mF25GnnvuOUqWLEn9+vU9fileLY4cOUKjRo04ePAgAwYMoHbt2mzcuJEJEyawbNkyVqxYQaFChZKdc/PNN9O/f/9k+wICAgBTsiha1DR0f/vtUL766iuGDzcTOrz66qsUKFCA9977p4PkqVOneOyxxxgzZsxVnWHnO6qaL7YGDRqo5UHz5mbLg7dfskQVVKdPd5Z+586dSc9r1aqlVapUydqN87jBgwcroF999VWy/V999ZUCOmbMmGT7Ae3bt2+613zgAdUiRVRDQ8vpqFGjkvaPHDlSK1SokCztgAEDtEGDBhoXF3dF78PKGUC0OviOddR11rJ8Ydo0KFwY7rrLWfqqVat6NyCHYmJi6Ny5MyVKlKBEiRJ07dqVEydOcPDgQYKDg5k2zbtreC1ZsoSCBQvSI8Vw9+7duxMcHMzkyZM9nhcbG8vZs2c9HuvZE86cgTNnLlCyZMmk/SVLlkxWDbV8+XI+/fRTPv74Y/z9/bPh3Vi5hZNqKMvKcRcvwowZcPfdJsPIK6ZNm0bfvn2pU6cOo0aNYvfu3fyR+Pv/AAAgAElEQVTnP/8hLCyMuLg4IiMj6e5hGHpCQgInTjifK7NkyZL4+Xn+rXfp0iWCg4MRSb4sjJ+fHwULFmTXrl0cO3aM0qVLJx2bOXMmX3zxBfHx8ZQpU4bu3bvz0ksvUaxYMcCMtyhXDhISmvDhhx/SvHlzVJUJEybQtGlTwGQ2Dz30EE8++ST16tVz/F6svCHDzEJE/FXVNmhbOeq//4WTJ6FPH19H4tyuXbvo168fdevWZdmyZQQHBwMQHR3N/Pnz2bNnD19++aXHL/l9+/YRkYl1Ynfv3p1me0CtWrXYunUr69ato27dukn7161bx99//510v8TMomHDhnTr1o1q1apx+vRp5s2bx/vvv89PP/3EL7/8QkhICP7+Zl6u8ePHUaRI+6TrRkZGMm7cOABefvllYmNjGT16tOP3YeUdTkoWO0RkJjBZVf/IMLVlZYPPPvtnBLE3nTx5MunLzolBgwYlq4Zx9+6773Lx4kXee++9pIwCTPXYL7/8Qv369encubPHc8uVK8fChc4nRyhXrlyax5544gm+++477rnnHsaNG0ft2rXZtGkTTzzxBAEBAVy+fJnz588npV+9enWy8/v06UOdOnUYMWIE7777LiNGjABMr6hx42owZMgmmjQxXwU1a9YkICCAP/74g9dee425c+dSsGBBPvjgAz744APOnDlDhw4deOONNyhYsKDj92flQhk1agBFgIeAX4BVQH+gqJMGkZzcbAN3GvJgA/fhw6oFCqg++2zW7+u0gXv37t2JMyo72rZv357mtcLDw7VatWqp9t93330K6Ny5c7P+hjJpxowZWq5cuaS4/f39dcCAAdq5c2cFdP369emeHxsbq4GBgdqkSZOkfQkJqpGRqi1aJE+bkJCgTZs2TWoknz59ugYHB+uXX36py5Yt06pVq+rDDz+c3W/RyiY4bODOsGShqmcws8x+LCK3ANOAd1yljTGquiO7Mi7LAjMALC4Oevf2/r3Cw8MTfxRdkRMnTrBnzx7u9TB68PDhw9SqVYs77rgjzfPj4+M5evSo4/uVKVMm3Qbkbt26cffdd7NhwwbOnDlDjRo1KFu2LA0bNqRAgQJUq1Yt3esHBARQoUIFjh07lrRPxPybjBxp1uhOrAWbMGEC27dvZ84csxDmJ598QpcuXejZ0/SwHz58OI8//jjvv/9+mu0sVu7nZKU8fxHpICLfYsZavAVUBf6PFAsbeTi3rYhsFZEdIjLMw/FbROQ3EYkTka4pjsW7Vs9LWkHPujp89hnUr5+3Ft05dOgQAKVKlUq2f9myZSxcuDDV/pT2799P+fLlHW/79+/PMCZ/f3/q1q3LzTffTNmyZTl06BAxMTE0b9481TiLlC5evMiBAwcIDQ1Ntr9PH5NpTJ1qXv/5558MHz6ccePGJb3HAwcOUNlteuDKlStz8eLFZBmPlfc4abPYDiwB3lTVX9z2z3SVNDwSEX9gPNAaOACsEZE5mrzdYx9wPzDEwyUuqGpdD/utfGzTJli7FjLRjJArFHeNGly/fn3SvsTRzoDHUc7usrPNwpOEhAQGDRpEfHx8UhsEwPHjxz1mZM8//zxxcXG0b98+2f4qVeDWW01m8fzz8Oijj9K0adOkUgRAhQoV2LBhQ9LrDRs2EBgYmKz3lZX3OMks+qjqcvcdInKTqq5Q1UHpnNcQ2KGqu1znTAc6AkmZharucR1LyGzgVv70+edmhtmszAX1+eefs3fvXgCOHj1KbGwsL71k5rysUqUKvb1Yr1WhQgUaNmzIzz//TO/evWnatCkff/wxx44do23btvz444+MGzeO7t27U758+VTnBwcH06pVmqsMZ8rZs2dp2LAhnTt3JiIiglOnTjFt2jTWrl3Lyy+/TEu3XgMvvfQSq1atomXLloSFhXH27FnmzZvHkiVLaNSoEY8//niq699/v6mOGjNmFosWLWLjxo3Jjt93333069ePJ554gkqVKjFmzBh69uxpq6DyuowaNYDfnOzzkKYrMMntdW/g/TTSTgG6ptgXB0RjGtU7pXFef1ea6LCwsGxs8slH8lAD9+XLquXKqbZvn9V7NU+zYbp5DvwN9uzZo3fddZeGhIRocHCwNmnSRDdv3qybN2/WmjVrKqBbtmzxehyXLl3S7t27a3h4uAYFBWmJEiW0TZs2+sMPP6RK+91332mbNm20QoUKGhQUpIUKFdIbbrhBX375Zb1w4YLH6587pxoSclILFiyvb7/9dqrjCQkJ+sorr2jlypW1ZMmS2qdPHz116lS2v08re+CwgVs0jcY9EWkCNAWeAN5xO1QU6KyqN6SXCYlIN8yMtf9yve4NNFTVVD9VRGQK8F9Vnem2r4KqHhSRqsD/gNtUdWda94uKitJoO+l+ai1amMelS3P97efMgY4d4bvvzKOVez30kJm2/NAhKFLE19FYV0JE1qpqVEbp0isXBgIhmKqqIm7baUypISMHAPdFMCsBBx2cB4CqHnQ97gKWAnZIaD73ySdmlHA6nYasXOKBB+D8eZg5M+O0Vv6QZpuFqv4E/CQiU1R1bxauvQaIFJEI4E+gBw5nqxWREsB5Vb0kIqWBm4A3shCDlUf89RfMnQvPPAOuyU6tXKxJE6heHSZPNhmHlf+lt6xqYn+U90VkTsotowurahzwGLAA2AzMUNVNIvKiiHRw3eNGETkAdAM+EpFNrtOvA6JFZD2mJ9ZrakeP52tTp5opyfv183UklhMiJpP4+WfYssXX0Vg5Ib3eUJ+7Hsdm9eKqOo8UYzFUdaTb8zWY6qmU5/0CXJ/V+1p5i6qpgmreHCIjfR2N5dQDD5jusxMnwttv+zoay9vSq4Za63r8KefCsa5Gy5bBjh1mZLCVd4SGQufOplT4yivgNh2WlQ+lVw21QUR+T2vLySCt/O3jj6FYMehiV3bPcwYMgBMnbEP31SC9aiiHS85YVtYdPQrffGO6YmYwA4WVC7VsCddcAx99BPfd5+toLG9KrxoqKz2gLCtTPv0UYmPhkUd8HYmVFX5+0L8/DB0Kf/wBNWv6OiLLW9KrhlruejwjIqdTPuZciFZ+FR8PEyaYX6f2Sybvuv9+09154kRfR2J5U5qZhao2cz0WUdWiKR9zLkQrv5o/H/butaWKvK5sWbP87dSpkMF8iVYe5mhmLxGpLyKDRORxEbEjqa1s8cEHZjU8O7VH3vfoo2YZ3C++8HUklrc4Wc9iJDAVKAWUBqaIyL+9HZiVv+3cCT/8YOq77YjtvK9ZM6hXD/7zHzNuxsp/nJQs7gVuVNVRqjoKaAz08m5Y3jF69GhEJGkrV64cd911F7//nrWewCLC+++/n60xbtu2jdGjR3Py5Mlk+6dMmYKIcPbs2Wy9n698+KGZivyhh3wdiZUdRGDQINPI/b//+ToayxucZBZ7APfhNkFAmrO/5nbFihVj5cqVrFy5knHjxrFt2zZat27NiRMnMn2tlStX0q1bt2yNb9u2bbzwwgupMos777yTlStXZrjCWV5w9ixMmmQGdFWo4OtorOzSoweUKQPvvuvrSCxvSLPrrIi8h1kL4BKwSUQWul63BpandV5uV6BAARo3bgxA48aNCQ8Pp0mTJvzwww/JVvtyIvE6OaFMmTKUKVMmx+6XVRcuXKBgwYLpppk82dRvP/VUDgVl5YjgYDNI7+WXTTXjNdf4OiIrO6VXsogG1gLfAs9hJvRbCowA5ns9shxyww1mWY6UaxqfOHGCAQMGEBoaSnBwME2bNmX16tXJ0niqhvr++++JiooiODiYcuXK8eyzz3L58uVkaX7//Xfat29P8eLFCQkJoWHDhixcuJClS5cmLWMZERGBiBAeHg54roY6duwYffv2pVSpUhQqVIgWLVqQck2P8FWrGLJzJ++88w6VKlWiRIkS9OjRI1XJJaUtW7bQo0cPKleuTKFChahVqxbjxo0jIeGfRQ2XLl2KiLBgwQI6dOhASEgIjz32GGCW8XzttddYvboay5YFUb16daZOnUp8vFkytWlTOH58Lq1bt6Zs2bIULVqUxo0b8+OPP6Ybl5W7PfywqV7M5tpZKxdIb1De1JwMxFf27dsHmC/nRJcuXaJVq1acPHmSN998k7JlyzJhwgRatWrF9u3b01z/eMaMGdx7770MGDCAV155hZ07dzJ8+HASEhIYO9bMx7hlyxZuuukmatSowYcffkipUqWIjo5m//79dO3albFjxzJkyBBmz55N+fLlCQoKSjP2Tp06sWPHDsaOHUvp0qV58803admyJTExMVSrVu2fuI4epc7ixUycOJEDBw7w1FNP8dxzz/HBBx+kee0///yTGjVq0KtXL4oUKcK6desYNWoUFy5cYPjw4cnSPvjggzzwwAM88cQTBLsmCHr88ceZOnUq5cuPpEiR+rRtu5B+/fqxZUspdu26izffhN27d9O+fXuGDBmCn58f8+fPp127dixbtoybbropg385KzeqUAG6dTODLV94AYraTvb5R0ZL6QGRwEzM2tm7Ejcny/Dl5NagQYMMlw8cNWqUlipVSi9fvqyXL1/WHTt2aKtWrbRu3bp68eLFpHSTJk3SgIAA3bZtW9K+y5cva9WqVXXIkCFJ+wB97733VNUsJRkWFqb3339/snt+8sknGhwcrMeOHVNV1R49emjFihX1/PnzHmP8v//7PwV09+7dyfZPnjxZAT1z5oyqqs6fP18BXbp0aVKas2fPaunSpbV///5J+6oEBWnV4GC9fPly0r7BgwdraGhohn+vRAkJCXr58mV9+eWXNSIiImn/kiVLFNAnnngiWfrt27eriOiUKVOSLavau3dvLVw4SqtWVY2LS36P+Ph4vXz5srZp00YfeOABx7FZuc+aNaqg+sYbvo7EcgKHy6o6aeCeDEzArIndEviMf6Yvz3OOHz9OQEAAAQEBVKtWjZiYGGbPnp3sF/yiRYto0KABERERxMXFERcXB0Dz5s1TVfMk2rZtG/v27eOee+5JOicuLo5bb72VixcvJi1q/7///Y/u3btnWK+fkV9//ZUyZcrQvHnzpH2FCxfmrrvuYvny5E1KLYsXp0CBfwqRNWvW5MiRI8TGxqZ5/YsXLzJq1CiqVatGUFAQAQEBjBgxgt27dyf9PRLdeeedyV4vXrwYPz8/OnfujGocquZvUaXKbZw7t45Bg+Lx94cDBw7Qt29fKlasSIECBQgICODHH39k27ZtV/KnsXwsKgpuuw3eeQcuXfJ1NFZ2SW8iwUQFVXWxiIia+aJGi8jPwCgvx+YVxYoVY9GiRcTHx7N+/XqGDBlCz549WbFiBX5+Ju88duwYq1atIsDDAIBr0mi1O3bsGAB3pLEmaGKbyPHjxylfvvwVv4+//vqL0NDQVPtDQ0NT9ewqXiD5P3NgYCCqSmxsLIGBgR6vP3ToUCZNmsSoUaOoX78+xYsX5/vvv+ell17i4sWLhISEJLunu2PHjhEfH0+xYsWS9rn/Kdu2/YuEhAp06NCBM2fO8OKLL1KtWjUKFy7MyJEjOXLkiOO/g5U7DR0KbdrA55/Dv/7l62is7OAks7goIn7AdhF5DLNEalknFxeRtsC7gD8wSVVfS3H8FmAcUAfooaoz3Y71BRIH/72k2dSGUqBAAaKizNrkjRo1omDBgvTp04dvvvmG7t27A1CyZEmioqKYMGFCqvPTakMoWbIkABMnTqRevdSD3BPbREqVKsVff/11xe+jfPnyHr9UDx8+nBTLlfjmm294/PHHefbZZ5P2zZ0712NaEUn2umTJkhQoUIAVK1bw8MMmAx4+3NRl338/RESUZceOHcTExDB//nzatm2bdO6FCxeuOHbL91q1gvr14c03zSJJ/v6+jsi6Uk6qoZ4ACgGDgAZAb6BvRieJiD8wHmgH1ATuFZGU08XtA+4HvkpxbklMyaUR0BAY5VqXO9vdd9991KpVi9dffz1p32233caOHTsICwsjKioq2Xb99Z4X8KtRowYVK1Zkz549qc6JioqiVKlSSdeeMWMGFy9e9HidxF/6aR1P1KhRI44cOcKyZcuS9p0/f565c+fSrFmzTP0NPLlw4UKyjDE+Pp7p06c7OvfWW28lPj6eU6dOUaRIFEWKRDFnThSFCkXxxhtRBAYGJmUK7vfYu3cvK1asuOLYLd8TMaWLbdvgu+98HY2VHTIsWahZ+hRX6WKQqp5xeO2GwA5V3eU6fzrQEdNQnnjtPa5jCSnOvR1YqKonXMcXAm2BaQ7v7ZiI8Nxzz9GrVy8WL17MbbfdRp8+ffjwww9p0aIFQ4YMoWrVqhw/fpxff/2VcuXK8eSTT6a6jp+fH2+99Ra9e/fm9OnTtGvXjsDAQHbt2sV3333HzJkzKVSoEKNGjeLGG2/klltu4emnn6ZUqVLExMRQqlQp+vXrR40aNQD46KOP6NGjB4UKFfKYQd1+++3cdNNNdO/enddee41SpUoxduxYLly4wDPPPHPFf5fWrVszfvx4qlWrRsmSJRk/fjyXHFZA16hRg4EDB9KjRw+KFXuWwMAofv75IjfdtInhw7cxadIkrr32WipVqsTTTz/NmDFjOHPmDKNGjaJixYpXHLuVO3TpYsZavP66mWgwRQHUymsyagEHooANmJHce4D1QAMH53XFVD0lvu4NvJ9G2ilAV7fXQ4B/u71+Hhji4bz+mPEg0WFhYRm2+if2hkopLi5OIyMjtU2bNkn7Tp48qYMGDdJKlSppQECAVqxYUTt37qzLly9370Wg77//frJrzZs3T5s1a6aFChXSIkWK6A033KAjRoxI1htp/fr12q5dOw0JCdGQkBBt2LChLlq0KOn42LFjNSwsTP39/bVKlSqqmro3lKrqkSNHtHfv3lq8eHENDg7WW265RX/99ddk8VQJCtKnK1VKts/TtVI6dOiQdurUSYsUKaJly5bVZ555RidOnJjsvMTeUBs2bEh1fkJCgr7zzjtaqFBNhUCF0tq48S06derUpDS//vqr3njjjRocHKzVqlXTyZMna9++fdVJzzYrb/joI9Mzav58X0dipQWHvaFEM5j1y7WE6qOq+rPrdTPgA1Wtk8F53YDbVfVfrte9gYaq+riHtFOA/6qrzUJEngGCVPUl1+vngfOq+lZa94uKitK0eip5w5kzZyhatChfffUV9957b47dN9NatDCPS5f65PZNmsDq1WYacjtQ6+oTGwvVq5tpzFevtqWL3EhE1qpqVEbpnLRZnEnMKABUdTngpCrqAFDZ7XUl4KCD8670XK/bsmULr776KiJC/fr1fR1OruYa88jQob6Nw/KNwEB4/nlYswbS6B9h5RHprZRXX0TqA7+KyEci0kJEmovIB5hpPzKyBogUkQgRCQR6AHMcxrUAaCMiJVwN221c+3KF4cOHM3XqVN58882kNgYrtb174a+/oFQpqFw54/RW/tSnj2m7GDnSTl+el6XXwJ2yysd9XEWG/+SqGufqarsA03X2U1XdJCIvYurI5ojIjZi5p0oA7UXkBVWtpaonRGQMJsMBeFFdjd25wbfffuvrEPKEUaPMl4P9gri6BQSYjKJvX9MzqnNnX0dkZUWGbRZ5RU63WeQZPmqz2LgR6tQx1RAlS8LBXFOJaPlCXBzUqmU+D+vXg5+jNTqtnJBtbRYiUkxE3haRaNf2logUy+g86+o2YgQEBZkvhwJOhn5a+VqBAmZiwY0b7dKreZWT/P1TTIP2Pa7tNGa+KMvy6JdfYM4cCAszGYZlAdxzj5k36rnn4Px5X0djZZaTzOIaNUuq7nJtLwBVvR2YlTepwrBhUKQIRETYrpLWP/z84O234c8/zaOVtzjJLC64xlYAICI3AXYCH8ujmTPh559N7yfbA8pK6eabTQP3a6/BoUO+jsbKDCeZxUBgvIjsEZE9wPvAAK9GZeVJ587B009D6dIQGWkbMS3PXn/dTF0+cqSvI7EyI93/zq75oGqo6g2YmWHrqGo9Vf09R6Kz8pTXX4f9+6FqVTNi17I8iYyExx6DTz6Bdet8HY3lVLqZhaomAI+5np9W1dM5EpWV5+zaBW+8AVWqQLVqtq3CSt/IkWaw5oABEB/v62gsJ5xUFCwUkSEiUllESiZuXo/MylOefto8RkSA27pIluVRiRKmkfvXX2HiRF9HYznhJLPoBzwKLAPWujY7+s1K8t13ZgsPN5mFZTnRqxfceqtZGMs2dud+GWYWqhrhYbNdZy0A/v4bHn4YypQxs4t6WInWsjwSgQ8+gAsX4KmnfB2NlREnI7iDReQpEZktIrNE5AkRCc6J4Kzc7+mn4cgRM1FcuXK+jsbKa2rUMCWLadNg3jxfR2Olx0k11GdALeA9TLfZmsDn3gzKyht+/BEmTzYZRWSkbdS2smb4cKhdGx58EI4f93U0VlqcZBY1VPVBVV3i2voD1b0dmJW7nToF/fubHi2RkVCwoK8jsvKqoCD4/HOTUTz8sJ2lOLdyklnEiEjjxBci0ghY4b2QrNxOFQYONGMqrrkG7LLZ1pWqW9dMNPjNN6ZKysp9nGQWjYBf3EZwrwSai8gG15Kr1lVmyhSYPt2UKK691lY/Wdnj2WehaVN49FHzQ8TKXZxMHt3W61FYecaWLWb0bYUKJqMItl0drGzi7w9Tp0L9+maG2p9+MlPcW7mDk66ze9Pb0jtXRNqKyFYR2SEiwzwcDxKRr13HV4tIuGt/uIhcEJF1ru3DrL5BK/tcuAD33mtKEtWrQ2ioryOy8ptq1eDTT2HVKhgyxNfRWO68NtWbiPgD44F2mB5U94pIzRTJHgT+VtVqwDvA627HdqpqXdc20FtxWs6owkMPmbl8IiNNW4VleUPXrmbcxXvv2faL3MSb84I2BHa41sCIBaYDHVOk6QhMdT2fCdwmYmvAc6M334QvvzT94mvXNlUGluUtr70GzZrBv/5lVtezfM+bmUVFwL2Z6oBrn8c0qhoHnAJKuY5FiEiMiPwkIjd7uoGI9E9c7vXo0aPZG72VZO5cs6BRlSpmHeVChXwdkZXfBQTA119DsWJw553w11++jsjyZmbhqYSQsgd1Wmn+AsJUtR7wFPCViBRNlVB1oqpGqWpUmTJlrjhgK7XffzftFGXKwHXXmbUqLCsnVKhgfqgcPw533QVnz/o6oqubNzOLA4D7WmmVgINppRGRAkAx4ISqXlLV4wCquhbYiR0ImON27IA2bUyD9nXX2ZXvrJxXrx7MmAHr10P37hAX5+uIrl7ezCzWAJEiEiEigUAPYE6KNHOAvq7nXYH/qaqKSBlXAzkiUhWIBHZ5MVYrhYMHoXVr82uudm07nYflO3fcAePHm7mj+vWz61/4ipNxFlmiqnEi8hiwAPAHPlXVTSLyIhCtqnOAT4DPRWQHcAKToQDcArwoInFAPDBQVU94K1YruWPH4PbbzbTRdeqYUoVdItXypQEDzOfy3/82nSs++cR+JnOa1zILAFWdB8xLsW+k2/OLQDcP580CZnkzNsuzv/6CVq1g+3aTUdieT1ZuMWKEqYYaPdp8JidOtBlGTvJqZmHlLXv3wm23wZ9/moyiTh0oYD8hVi4yapSphhozBmJjYdIkO8o7p9ivAguAP/4wVU8nTphMolYtu5CRlTu98IKZqfbf/zZVpTNnQtFUfSWt7GYLcRZz50Ljxmba8cSqJ5tRWLmViKmSmjwZliyB5s1NhwzLu2xmcRVThbFjoX17sx5F/fomo7BVT1ZecP/98N//mi7eDRrAsmW+jih/s5nFVervv02/9WeeMeMn6tUzkwPaBkMrL7n9dli50lRD3XorvPWWXTzJW+xXw1Vo+XKz2Mzs2WaupxtvNFN52HEUVl5UuzasWQMdO5qZajt0sNODeIPNLK4i587B0KGmjvfMGVN0b9TILI1qWXlZ0aKmoXvcOFi0yHTQ+OorW8rITjazuErMn29+gb3xhql2atgQbrjBLl5k5R8iMHiwmUa/Rg3o1cuUMnbs8HVk+YPNLPK5c+dhw0YzZcKZMxAVZaZ+ttVOVn5Vo4apah07FpYuNaWMYcPM59/KOptZ5FP79pm1ANasgb9PmMbrxo1NQ3bhwr6OzrK8y98fnn4atm0zsya//jpUrWpK1nb22qyxmUU+s2ED9OljVrKbOhUKBps1AZo1g4oVr6w0ER09mokTJdn2+eflkqXZvv1LvvyyMlOnlmTlyqeSHfvzzz8JDw/n8OHDWQ/CsjKhfHmYMgVWrzal6qFDISICXnnFzDVlOWd71OcDsbHw/fdmrpxFi8zo1sqVTVVTsc0gftk3dqJYsRq0b7806bVrcmAALl48xrJl/6JFiykUKVKVH364k8DAWwkOvguARx99lOeff55Qu3i3lcMaNjTtdqtWmRHgI0aYKUN69oRHHzUlblstmz5bssijVM0H/8knoVIluOeetSxZchsiBfHzq0aNGsvw959BzxMx2XpfP78CFCpULmkrWPCfRadOn95FYGAxrrmmO2XL3kiFCi2Jj98MwKxZszh16hT9+vXL1ngsKzMaNzaZxsaNZlDf9OmmV2CdOmbp4AMHfB1h7mVLFnnIuXPw00+wYAF8951plyhQAIoXX4OfX3OqV3+O2rU/JCbmBdavH8WlSyd4u0h4smvExLxCTMwr6d6nXbv5lC/vcSVbTp/exRdfVMTfP5AyZRrRsOErFC1aFYBixSKJizvPsWMxhIRU4ejRNRQo0I+EhFM888wz/PDDD9gl1q3coFYtmDABXn3VLN/62Wfw7LNmq1/frMzXvr15bgeqGqL5pCNyVFSURkdH+zqMbHXhAvz2G6xYYTKI5ctNlVNAgBkbUbYshIbC+vW3EBJSnlatvgZg3755LFjQnnLlbmGJayXbt11VRxcvnuDSpfSXBilcuCIFChRMtX/fvvlcvnyG4sWv5cKFI8TEvMTJk1vo1m0TwcFmsMbu3d8SHT2S+PgLVKt2H/v3j+b8+QE891wVWrRowaBBgzh37g90g38AAAzASURBVByDBw9m4MCB2fjXsqwrs327Gaj6f/9nRoUnJJg2j1tvhZtuMu1+tWrlv8xDRNaqalRG6WzJIpc4dQo2b4ZNmyA6Gn791ax/nbiMZMmSpoG6dGmzHnaZMqZt4vz5Qxw58jMNGy5JupafXyCqCdx44xhY8+9k9wkOLklwcMksxRgW1i7Z69DQxkybVpVt26ZSp45pzI6I6ExEROekNLt2LSc2dhWDBr1FjRo1mDp1KrVq1aJOnTrcdNNNXH/99VmKxbKyW2SkaQAfOtQ0fs+fbybZ/N//4MsvTZrixU1VVt26/0zjX7361THxplczCxFpC7yLWSlvkqq+luJ4EPAZ0AA4DnRX1T2uY8OBBzEr5Q1S1QXejNXbEhLMFAR79ybfdu0yGcSff/6TNijIfCirVDGPxYubzKFIkdSNcCdPmjaBMmVuTNp36tRWihWrQblyzVLFcaXVUO4CAkIoUaIWp05t93g8Pj6WM2cGUqrUJHbt2kVsbCytWrUCoEWLFixdutRmFlauVLo09O5tNlXYvduU7FesMG2FixfD5csmbWAgVKtmuuam3CpX9vz/Ni/yWmbhWkN7PNAaOACsEZE5qvqHW7IHgb9VtZqI9ABeB7qLSE3MEqu1gArAIhGprqo+X303IcH00z592pQGTp9O/vzECThyBI4e/efx8GHzPDY2+bWCg6FQIfNYowaEhJgxEKVLmw9YYGDGH7JLl04CgogpG8fGniEm5mUKFiznMf111w2katV70r1m4cIVHf0t4uIucvLkFipUaOnxeEzMywQG3kpQUGMSEtYRl1hMAmJjY4m3iylbeYDIP1/+ffqYfbGxsHWrKf3//rupwtq1ywwCTDmOo2BBKFfOVBmXK2e2MmVMl/bEH4MpnxcubL4XclOVlzdLFg2BHaq6C0BEpgMdAffMoiMw2vV8JvC+mBbQjsB0Vb0E7Hat0d0QWJndQZ48CY88Ahcv/rNduJD80X27dCnjaxYoYP6hAwPN4KCAAPMLo2BBsz8oyHwgQkL+eZ3VXx6lStUFlHXrXqVatV6sWvUMhQqV59SpHR5/8V9JNdSqVUMIC2tPSEgYFy8e4bffxhAXd47q1fumSvv333+wY8eXFCpkemPVqFGDAgUK8OGHH1KrVi0WL17M888/n6U4LMvXAgPh+uvN1qvXP/tV4fhxk3Hs3Gl6Vx0+bBZpOnTITD2yfLlJ46S5ODjYfG8ULGh+WLo/T/yOCQyEa681XYG9yZuZRUVgv9vrA0CjtNKoapyInAJKufavSnFuqp+7ItIf6A8QFhaWpSBVTWPW33+bL2w/P7P5+//zukAB80u/aNHkady/4EVMuqAgsyX+IwYEeP51oGqmH7jyKQgiqF79ZTZseId1694g9P/bu/sYuao6jOPfh0JpQ7ElFN/Ypq22AaHSBRFQjEJFsmKtAW0EI5Co0cTWIPUFYRPwJRVIVfwDAoIQTMGXqhAbwJSSggS0hZbXNmWlaLSrIkjtWgq11v7849xJp+Pu3l12Z87szPNJJrN39t65T7ad+d17zrn3vOGjdHbex/r187njjndz2aHHArB1a8nbDMGLL/bS03Meu3f/g/Hjj2DKlFM4+eS1bN8+ne3b960XEaxb91lmzbqGLVsOJQImTpzI8uXLWbRoEX19fXR3d3PiiaV9amZjipRaBqZOTdd2DGTv3vTZ7+tLB6y1zzt3poPVV1+FV17Z/7ny87Zt6Qxn9+7GnIHUs1j0d6xcW0sHWmco2xIRNwI3QhoNNdyAAIcdlq56fuCBtDxuXPrSHzdu0M2azGXFo9rDAExbchqQbt08cj8d4noCHgJgyZJ9cyR3dXXx3HPPjUYQszHtgANS68LkyfAaj3Mbrp7FoheYVrXcAdROflhZp1fSgcBkYNsQtx01kyalcdUtaUp6mjcv0+6n5NmvmY2uep68PArMljRT0nhSh/XKmnVWApUG748BayJd+LESOFfSwZJmArOBR+qY1czMBlG3M4uiD2IxsIo0dPaWiNgk6ZvA+ohYCdwMLC86sLeRCgrFeitIneF7gEXNMBLKzKxd1fU6i4i4B7in5rXLq37eBSwcYNulwNJ65jMzs6FpolG8ZmbWrFwszMyslIuFmZmVcrEwM7NSLXOLckkvAn8awVtMBZpxokXnGh7nGh7nGp5WzDU9Io4oW6llisVISVo/lHu6N5pzDY9zDY9zDU8753IzlJmZlXKxMDOzUi4W+9yYO8AAnGt4nGt4nGt42jaX+yzMzKyUzyzMzKyUi4WZmZVysagh6cuSQtLU3FkqJH1L0lOSnpB0r6Q3N0GmZZKeKXLdKalpZq6QtFDSJkl7JWUd5iipS1KPpC2SvpYzSzVJt0h6QdLG3FmqSZom6X5Jm4t/w4tyZwKQNEHSI5KeLHJ9I3emCknjJD0u6a567sfFooqkacAHgD/nzlJjWUQcFxGdwF3A5WUbVJN0saSLRznTamBORBwH/B64dJTffyQ2AucAD+YMIWkccB3wQeAY4DxJx+TMVOVWoCt3iH7sAb4UEW8DTgEWNcnf7N/AvIiYC3QCXZJOyZyp4iJgc7134mKxv2uAr9LPFK45RcS/qhYPYRj5JH0OuBK4spizfLQy3RsRe4rFtaTZDJtCRGyOiJ7cOYCTgC0R8YeI2E2al/YjmTMBEBEPkuaQaSoR8beIeKz4eQfpS/DIvKkgkpeLxYOKR/bvCUkdwIeAH9Z7X3Wdz2IskbQA+EtEPCn1NwV4XpKWAhcAfcDpQ9zmLcB3gC+QDgy+K2l1RPxxlON9CvjZKL9nKzgS2Fq13AucnCnLmCNpBnA8sC5vkqQ4U9wAzAKui4hmyPV90gHuofXeUVsVC0n3AW/s51fdwGXAmY1NtM9g2SLiVxHRDXRLuhRYDFxR8n4HkJoafhERNxWvnQrcKun0iNg70kzFOt2kpoPby95vNA0lWxPo76gj+9HoWCBpEvBL4Is1Z9bZFLN1dhb9c3dKmhMR2fp8JM0HXoiIDZJOq/f+2qpYRMQZ/b0u6e3ATKByVtEBPCbppIh4Pme2fvwYuJuSYlEUg/fWvHbBaGaSdCEwH3h/NPiCnWH8vXLqBaZVLXcAf82UZcyQdBCpUNweEXfkzlMrIrZLeoDU55NzgMCpwAJJZwETgNdJui0iPlmPnbnPAoiIpyPi9RExIyJmkD7kJzSqUJSRNLtqcQHwTK4sFZK6gEuABRHxSu48TepRYLakmZLGk+aYX5k5U1NTOlq7GdgcEd/LnadC0hGVEX+SJgJnkPlzGBGXRkRH8Z11LrCmXoUCXCzGiqskbZT0FKmprBmGE15LaiddXQzpvSF3oApJZ0vqBd4F3C1pVY4cxQCAxcAqUkftiojYlCNLLUk/AX4HHCWpV9Knc2cqnAqcD8wr/l89URw55/Ym4P7iM/gosDoi6jpUtdn4dh9mZlbKZxZtQNKS4kLDT+TOYmZjk4tFeziheN6QNYWZjVluhmoDkjaTRuJMHsqQWTOzWj6zaGGSrpYUwNHAJOC/RXNUSKrbqAkzaz1tdZ1FG9oA/Ai4EPgt6X5OFb/JksjMxiSfWbSwiFhBKhIAt0bE16seWwfbtp7aqcNd0hRJn8+dw2ykXCxaX6Vz+/GsKfbXTh3uUwAXCxvzXCxa3wnAf4Cncwep8g7gZeDZ3EEa4CrgrcXFZctyhzF7rTwaqoVJOhDYAfQUc2HkznM16Q6Z/Tk/Im5rZJ5GKO6celdEzMkcxWxE3MHd2o4h3WDssdxBCu5wNxujXCxaW+Vsoin6KyJiRXEztgtJHe435c5kZkPjPovWdnjx3BTzARSascO9nnbQgIlpzOrNZxatrTLaaKmkOcBOYFNE/DxjpmbscK+biHhJ0sOSNgK/joiv5M5k9lq4g7vFSVpMmlZ1OnAw8O1i1r0cWUa1w13SgVXzgJtZHbkZqsVFxLURcVRETIgI5SoUhUE73CUdLul6Sc9L2lXM4XFm8buO4kK+cyWtkbSLNCe5mTWAm6GskQbscJfUAawF1gPnAC8A72Fff0tl20tIc6b3ANvrGdbM9nGxsEYarMP9B8BTwNlV83lvqfr9XGAXsDAittRubGb15WJhjdRvhzuwDjgLeGcM3InWCdzjQmGWh/ssrGEi4kFSZ/vO4vkKUhE4HtjD4PeKmosv3DPLxqOhLDtJZwF3kyZn+r8mKkmHkJqu3hcRDzU6n5n5zMKawzrgn8ANko6VdLSkz0iaW/z+uOL5iTzxzMzFwrKLiJeAD5OuBVlbPD4O/L1YZS7wbES8nCehmbkZyszMSvnMwszMSrlYmJlZKRcLMzMr5WJhZmalXCzMzKyUi4WZmZVysTAzs1IuFmZmVup/W3qONpXsNlAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "df = n_samples-1\n",
    "\n",
    "x = np.linspace(-4,4, 200)\n",
    "w1 = t.ppf(0.05,df)\n",
    "x1 = np.linspace(-4,w1, 200)\n",
    "w2 = t.ppf(0.95,df)\n",
    "x2 = np.linspace(w2,4, 200)\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "\n",
    "y = t.pdf(x,df)\n",
    "y1 = t.pdf(x1,df)\n",
    "#y2 = t.pdf(x2,df)\n",
    "\n",
    "ax.plot(x, y, 'b-')\n",
    "plt.fill_between(x1,y1, color = 'b',alpha=0.6)\n",
    "#plt.fill_between(x2,y2, color = 'b')\n",
    "\n",
    "plt.xlabel('t')\n",
    "plt.ylabel('probability density function')\n",
    "\n",
    "plt.text(-1.1, 0.15, r'$1-\\alpha = 95\\%$', fontsize=18)\n",
    "plt.text(-2.8, 0.0, r'$\\alpha = 5\\%$', fontsize=14)\n",
    "#plt.text(2.1, 0.1, r'$\\alpha/2 = 5\\%$', fontsize=18)\n",
    "plt.text(-4.2, 0.1, r'Rejection area', fontsize=15)\n",
    "t_crit =  np.round(t.ppf(0.05,df),3)\n",
    "plt.text(t_crit, -0.09, r'$t_{cr}$', fontsize=20)\n",
    "plt.text(t_stat-0.1, -0.08, r'$\\hat t$', fontsize=20)\n",
    "plt.text(0.5, 0.35, r'$H_0|n=25$', fontsize=20)\n",
    "\n",
    "\n",
    "plt.axvline(x=t_stat, color='r')\n",
    "plt.axvline(x=t_crit, color='b')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p:  0.88 %\n"
     ]
    }
   ],
   "source": [
    "print(\"p: \",np.round(t.cdf(t_stat,df)*100,2),\"%\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "#from ipywidgets import *\n",
    "#import numpy as np\n",
    "#import matplotlib.pyplot as plt\n",
    "# %matplotlib inline\n",
    "# from scipy.stats import norm\n",
    "# from scipy.stats import t\n",
    "\n",
    "# x = np.linspace(-5,5,100)\n",
    "# p_0 = 0\n",
    "# s_0 = 1\n",
    "\n",
    "# def update(n = (1,40,2)):\n",
    "#     fig = plt.figure()\n",
    "#     ax = fig.add_subplot(1, 1, 1)\n",
    "#     ax.fill_between(x, t.pdf(x,n))\n",
    "#     ax.plot(x, norm.pdf(x),color='k')\n",
    "#     plt.xlim(-5,5)\n",
    "#     plt.ylim(0,0.42)\n",
    "#     plt.xlabel('x')\n",
    "#     plt.ylabel('probability density')\n",
    "\n",
    "#     fig.canvas.draw()\n",
    "# interact(update);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "![TvsN.gif](https://raw.githubusercontent.com/antonfadic/DSproject/master/TvsN.gif)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p:  0.54 %\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "print(\"p: \",np.round(norm.cdf(t_stat)*100,2),\"%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# On the normality assumption\n",
    "### Convergence to normality\n",
    "In the previous plot it is shown that the distribution converges to the normal distribution as the sample gets larger. Therefore, for large samples, it is safe to use a normal distribution as an approximation of the t distribution. \n",
    "\n",
    "Other cases of convergence to the normal distribution exist. For instance, to compute the average of a binomial process, the average is distributed normal for a large enough distribution. The reason for this is the __Central Limit Theorem__, which in its more well known form says that the normalized sum of random variables identically distributed, uncorrelated and with finite variance, converges in distribution to a normal distribution. Which, in more plain English, is saying that given certain circumstances, the __sum__ of random variables is distributed normal. When we want to know a mean, we use the sum, and this is why it can be useful for inference.\n",
    "\n",
    "### Testing for normality,\n",
    "Before you do any of the tests we performed in this notebook, it is adviced to check your assumption of normality with a test. In particular you can use __Anderson-Darling test, Kolgomorov-Smirnov test__ among others. It is important to see the what is the recommended usage of any of these tests, since sometimes the results can be contradicting.\n",
    "\n",
    "### What to do if the distribution is not normal?\n",
    "If we are testing for averages, it is safe to say that for a large enough sample, usually taken to be >30 or 50 (be careful with this) that the distribution of the average is normal. However, if the samples are small, then this assumption cannot be made. Here other tests can be used, in particular: __non-parametric tests__. An example of it is __Kruskal-Wallis test__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Setting the level of confidence. Precision and recall\n",
    "\n",
    "* The level of confidence affects the decision of rejecting or non-rejecting the null hypothesis\n",
    "* Rejecting the null hypothesis, when it is true, is called __type I error__. This is a __false positive, or false alarm__\n",
    "* Non-rejecting the null hypothesis, when it is false, is called __type II error__ . This is a __miss or fail__\n",
    "\n",
    "__Type I errors and Type II errors can have very different consequences!__ Two examples:\n",
    "\n",
    "* Illness detection: prefer false positives. You don't want to be a false negative!\n",
    "* Spam detection: prefer against false positives. You don't want to miss important emails!\n",
    "\n",
    "__Precision__ \n",
    "\n",
    "$pr = \\frac{tp}{tp+fp}$ \n",
    "\n",
    "\n",
    "__Recall__ \n",
    "\n",
    "$re = \\frac{tp}{tp+fn}$\n",
    "\n",
    "One way to combine both measures of effectivness of the test is called the __F$_1$ score__, computed as:\n",
    "\n",
    "$F_1 = \\frac{2}{\\frac{1}{pr} + \\frac{1}{re}}$ \n",
    "\n",
    "Another way to combine both of them:\n",
    "\n",
    "$F_{\\beta} = (1+\\beta^2) \\frac{1}{\\frac{\\beta^2}{re} + \\frac{1}{pr}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Summary\n",
    "\n",
    "When doing hypothesis testing, check the following.\n",
    "\n",
    "* Do you know the distribution of your data? Is it normal? Do you know the population variance?\n",
    "* If you don't know the variance, but you know that the data comes from a normal, you can use the student t distribution with n-1 degrees of freedom. You can approximate the population variance by the sample variance.\n",
    "* To do hypothesis testing you need: A null hypothesis $H_0$, and alternative hypothesis $H_a$, a level of confidence $1-\\alpha$, data, and a test statistics $\\hat t$. \n",
    "* Be aware of type I and type II errors, and how they relate to the level of confidence."
   ]
  }
 ],
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