Advanced Statistical Methods in Python - StatsModel.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import scipy\n",
    "import statsmodels.api as sm\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import sklearn\n",
    "sns.set()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
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       "\n",
       "    .dataframe thead th {\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SAT</th>\n",
       "      <th>GPA</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1714</td>\n",
       "      <td>2.40</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1664</td>\n",
       "      <td>2.52</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1760</td>\n",
       "      <td>2.54</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1685</td>\n",
       "      <td>2.74</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1693</td>\n",
       "      <td>2.83</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    SAT   GPA\n",
       "0  1714  2.40\n",
       "1  1664  2.52\n",
       "2  1760  2.54\n",
       "3  1685  2.74\n",
       "4  1693  2.83"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data =pd.read_csv(\"1.01. Simple linear regression.csv\")\n",
    "data.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SAT</th>\n",
       "      <th>GPA</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>84.000000</td>\n",
       "      <td>84.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>1845.273810</td>\n",
       "      <td>3.330238</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>104.530661</td>\n",
       "      <td>0.271617</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>1634.000000</td>\n",
       "      <td>2.400000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>1772.000000</td>\n",
       "      <td>3.190000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>1846.000000</td>\n",
       "      <td>3.380000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>1934.000000</td>\n",
       "      <td>3.502500</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>2050.000000</td>\n",
       "      <td>3.810000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               SAT        GPA\n",
       "count    84.000000  84.000000\n",
       "mean   1845.273810   3.330238\n",
       "std     104.530661   0.271617\n",
       "min    1634.000000   2.400000\n",
       "25%    1772.000000   3.190000\n",
       "50%    1846.000000   3.380000\n",
       "75%    1934.000000   3.502500\n",
       "max    2050.000000   3.810000"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Creating the first regression model\n",
    "\n",
    "Define the dependent and independent variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = data['GPA']\n",
    "x1 = data['SAT']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x1,y)\n",
    "plt.xlabel('SAT', fontsize = 20)\n",
    "plt.ylabel('GPA', fontsize = 20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Regression itself"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>           <td>GPA</td>       <th>  R-squared:         </th> <td>   0.406</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.399</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   56.05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sat, 09 Jan 2021</td> <th>  Prob (F-statistic):</th> <td>7.20e-11</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>22:34:49</td>     <th>  Log-Likelihood:    </th> <td>  12.672</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    84</td>      <th>  AIC:               </th> <td>  -21.34</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    82</td>      <th>  BIC:               </th> <td>  -16.48</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "    <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th> <td>    0.2750</td> <td>    0.409</td> <td>    0.673</td> <td> 0.503</td> <td>   -0.538</td> <td>    1.088</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>SAT</th>   <td>    0.0017</td> <td>    0.000</td> <td>    7.487</td> <td> 0.000</td> <td>    0.001</td> <td>    0.002</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>12.839</td> <th>  Durbin-Watson:     </th> <td>   0.950</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.002</td> <th>  Jarque-Bera (JB):  </th> <td>  16.155</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.722</td> <th>  Prob(JB):          </th> <td>0.000310</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 4.590</td> <th>  Cond. No.          </th> <td>3.29e+04</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 3.29e+04. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                    GPA   R-squared:                       0.406\n",
       "Model:                            OLS   Adj. R-squared:                  0.399\n",
       "Method:                 Least Squares   F-statistic:                     56.05\n",
       "Date:                Sat, 09 Jan 2021   Prob (F-statistic):           7.20e-11\n",
       "Time:                        22:34:49   Log-Likelihood:                 12.672\n",
       "No. Observations:                  84   AIC:                            -21.34\n",
       "Df Residuals:                      82   BIC:                            -16.48\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const          0.2750      0.409      0.673      0.503      -0.538       1.088\n",
       "SAT            0.0017      0.000      7.487      0.000       0.001       0.002\n",
       "==============================================================================\n",
       "Omnibus:                       12.839   Durbin-Watson:                   0.950\n",
       "Prob(Omnibus):                  0.002   Jarque-Bera (JB):               16.155\n",
       "Skew:                          -0.722   Prob(JB):                     0.000310\n",
       "Kurtosis:                       4.590   Cond. No.                     3.29e+04\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 3.29e+04. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sm.add_constant(x1)\n",
    "results = sm.OLS(y,x).fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x1,y)\n",
    "yhat = 0.2750 + 0.0017*x1\n",
    "fig = plt.plot(x1, yhat, lw=4, c=\"red\", label=\"regression line\")\n",
    "plt.xlabel('SAT', fontsize = 20)\n",
    "plt.ylabel('GPA', fontsize = 20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x1,y)\n",
    "yhat = 0 + 0.0017*x1\n",
    "fig = plt.plot(x1, yhat, lw=4, c=\"green\", label=\"regression line\")\n",
    "plt.xlabel('SAT', fontsize = 20)\n",
    "plt.ylabel('GPA', fontsize = 20)\n",
    "plt.xlim(0)\n",
    "plt.ylim(0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x1,y)\n",
    "yhat = 0 + 0*x1\n",
    "fig = plt.plot(x1, yhat, lw=4, c=\"yellow\", label=\"regression line\")\n",
    "plt.xlabel('SAT', fontsize = 20)\n",
    "plt.ylabel('GPA', fontsize = 20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multiple Linear Regression and Adjusted R-squared"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import scipy\n",
    "import statsmodels.api as sm\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import sklearn\n",
    "sns.set()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SAT</th>\n",
       "      <th>Rand 1,2,3</th>\n",
       "      <th>GPA</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1714</td>\n",
       "      <td>1</td>\n",
       "      <td>2.40</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1664</td>\n",
       "      <td>3</td>\n",
       "      <td>2.52</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1760</td>\n",
       "      <td>3</td>\n",
       "      <td>2.54</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1685</td>\n",
       "      <td>3</td>\n",
       "      <td>2.74</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1693</td>\n",
       "      <td>2</td>\n",
       "      <td>2.83</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    SAT  Rand 1,2,3   GPA\n",
       "0  1714           1  2.40\n",
       "1  1664           3  2.52\n",
       "2  1760           3  2.54\n",
       "3  1685           3  2.74\n",
       "4  1693           2  2.83"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data1 = pd.read_csv('1.02. Multiple linear regression.csv')\n",
    "data1.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
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       "\n",
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       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SAT</th>\n",
       "      <th>Rand 1,2,3</th>\n",
       "      <th>GPA</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>84.000000</td>\n",
       "      <td>84.000000</td>\n",
       "      <td>84.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>1845.273810</td>\n",
       "      <td>2.059524</td>\n",
       "      <td>3.330238</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>104.530661</td>\n",
       "      <td>0.855192</td>\n",
       "      <td>0.271617</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>1634.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>2.400000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>1772.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>3.190000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>1846.000000</td>\n",
       "      <td>2.000000</td>\n",
       "      <td>3.380000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>1934.000000</td>\n",
       "      <td>3.000000</td>\n",
       "      <td>3.502500</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>2050.000000</td>\n",
       "      <td>3.000000</td>\n",
       "      <td>3.810000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               SAT  Rand 1,2,3        GPA\n",
       "count    84.000000   84.000000  84.000000\n",
       "mean   1845.273810    2.059524   3.330238\n",
       "std     104.530661    0.855192   0.271617\n",
       "min    1634.000000    1.000000   2.400000\n",
       "25%    1772.000000    1.000000   3.190000\n",
       "50%    1846.000000    2.000000   3.380000\n",
       "75%    1934.000000    3.000000   3.502500\n",
       "max    2050.000000    3.000000   3.810000"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data1.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = data1['GPA']\n",
    "x1 = data1[['SAT', 'Rand 1,2,3']]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = sm.add_constant(x1)\n",
    "results = sm.OLS(y,x1).fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>           <td>GPA</td>       <th>  R-squared (uncentered):</th>      <td>   0.996</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared (uncentered):</th> <td>   0.996</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th>          <td>1.048e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sat, 09 Jan 2021</td> <th>  Prob (F-statistic):</th>          <td>1.69e-99</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>22:30:09</td>     <th>  Log-Likelihood:    </th>          <td>  12.459</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    84</td>      <th>  AIC:               </th>          <td>  -20.92</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    82</td>      <th>  BIC:               </th>          <td>  -16.06</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>              <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>              <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "       <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>SAT</th>        <td>    0.0018</td> <td> 3.22e-05</td> <td>   56.146</td> <td> 0.000</td> <td>    0.002</td> <td>    0.002</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Rand 1,2,3</th> <td>   -0.0051</td> <td>    0.027</td> <td>   -0.190</td> <td> 0.850</td> <td>   -0.058</td> <td>    0.048</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 9.821</td> <th>  Durbin-Watson:     </th> <td>   1.122</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.007</td> <th>  Jarque-Bera (JB):  </th> <td>  11.353</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.594</td> <th>  Prob(JB):          </th> <td> 0.00343</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 4.353</td> <th>  Cond. No.          </th> <td>2.15e+03</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] R² is computed without centering (uncentered) since the model does not contain a constant.<br/>[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[3] The condition number is large, 2.15e+03. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                                 OLS Regression Results                                \n",
       "=======================================================================================\n",
       "Dep. Variable:                    GPA   R-squared (uncentered):                   0.996\n",
       "Model:                            OLS   Adj. R-squared (uncentered):              0.996\n",
       "Method:                 Least Squares   F-statistic:                          1.048e+04\n",
       "Date:                Sat, 09 Jan 2021   Prob (F-statistic):                    1.69e-99\n",
       "Time:                        22:30:09   Log-Likelihood:                          12.459\n",
       "No. Observations:                  84   AIC:                                     -20.92\n",
       "Df Residuals:                      82   BIC:                                     -16.06\n",
       "Df Model:                           2                                                  \n",
       "Covariance Type:            nonrobust                                                  \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "SAT            0.0018   3.22e-05     56.146      0.000       0.002       0.002\n",
       "Rand 1,2,3    -0.0051      0.027     -0.190      0.850      -0.058       0.048\n",
       "==============================================================================\n",
       "Omnibus:                        9.821   Durbin-Watson:                   1.122\n",
       "Prob(Omnibus):                  0.007   Jarque-Bera (JB):               11.353\n",
       "Skew:                          -0.594   Prob(JB):                      0.00343\n",
       "Kurtosis:                       4.353   Cond. No.                     2.15e+03\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] R² is computed without centering (uncentered) since the model does not contain a constant.\n",
       "[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[3] The condition number is large, 2.15e+03. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Dealing with Categorical Data using Dummy Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import statsmodels.api as sm\n",
    "import seaborn as sns\n",
    "sns.set()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SAT</th>\n",
       "      <th>GPA</th>\n",
       "      <th>Attendance</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1714</td>\n",
       "      <td>2.40</td>\n",
       "      <td>No</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1664</td>\n",
       "      <td>2.52</td>\n",
       "      <td>No</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1760</td>\n",
       "      <td>2.54</td>\n",
       "      <td>No</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1685</td>\n",
       "      <td>2.74</td>\n",
       "      <td>No</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1693</td>\n",
       "      <td>2.83</td>\n",
       "      <td>No</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    SAT   GPA Attendance\n",
       "0  1714  2.40         No\n",
       "1  1664  2.52         No\n",
       "2  1760  2.54         No\n",
       "3  1685  2.74         No\n",
       "4  1693  2.83         No"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "raw_data = pd.read_csv(\"1.03. Dummies.csv\")\n",
    "raw_data.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "data2 = raw_data.copy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SAT</th>\n",
       "      <th>GPA</th>\n",
       "      <th>Attendance</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1714</td>\n",
       "      <td>2.40</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1664</td>\n",
       "      <td>2.52</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1760</td>\n",
       "      <td>2.54</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1685</td>\n",
       "      <td>2.74</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1693</td>\n",
       "      <td>2.83</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    SAT   GPA  Attendance\n",
       "0  1714  2.40           0\n",
       "1  1664  2.52           0\n",
       "2  1760  2.54           0\n",
       "3  1685  2.74           0\n",
       "4  1693  2.83           0"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data2[\"Attendance\"] = data2[\"Attendance\"].map({\"Yes\":1, \"No\":0})\n",
    "data2.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SAT</th>\n",
       "      <th>GPA</th>\n",
       "      <th>Attendance</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>84.000000</td>\n",
       "      <td>84.000000</td>\n",
       "      <td>84.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>1845.273810</td>\n",
       "      <td>3.330238</td>\n",
       "      <td>0.464286</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>104.530661</td>\n",
       "      <td>0.271617</td>\n",
       "      <td>0.501718</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>1634.000000</td>\n",
       "      <td>2.400000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>1772.000000</td>\n",
       "      <td>3.190000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>1846.000000</td>\n",
       "      <td>3.380000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>1934.000000</td>\n",
       "      <td>3.502500</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>2050.000000</td>\n",
       "      <td>3.810000</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               SAT        GPA  Attendance\n",
       "count    84.000000  84.000000   84.000000\n",
       "mean   1845.273810   3.330238    0.464286\n",
       "std     104.530661   0.271617    0.501718\n",
       "min    1634.000000   2.400000    0.000000\n",
       "25%    1772.000000   3.190000    0.000000\n",
       "50%    1846.000000   3.380000    0.000000\n",
       "75%    1934.000000   3.502500    1.000000\n",
       "max    2050.000000   3.810000    1.000000"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data2.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = data2[\"GPA\"]\n",
    "x1 = data2[[\"SAT\", \"Attendance\"]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sm.add_constant(x1)\n",
    "results = sm.OLS(y,x).fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>           <td>GPA</td>       <th>  R-squared:         </th> <td>   0.565</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.555</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   52.70</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 11 Jan 2021</td> <th>  Prob (F-statistic):</th> <td>2.19e-15</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>22:09:50</td>     <th>  Log-Likelihood:    </th> <td>  25.798</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    84</td>      <th>  AIC:               </th> <td>  -45.60</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    81</td>      <th>  BIC:               </th> <td>  -38.30</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "       <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th>      <td>    0.6439</td> <td>    0.358</td> <td>    1.797</td> <td> 0.076</td> <td>   -0.069</td> <td>    1.357</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>SAT</th>        <td>    0.0014</td> <td>    0.000</td> <td>    7.141</td> <td> 0.000</td> <td>    0.001</td> <td>    0.002</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Attendance</th> <td>    0.2226</td> <td>    0.041</td> <td>    5.451</td> <td> 0.000</td> <td>    0.141</td> <td>    0.304</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>19.560</td> <th>  Durbin-Watson:     </th> <td>   1.009</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td> <th>  Jarque-Bera (JB):  </th> <td>  27.189</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-1.028</td> <th>  Prob(JB):          </th> <td>1.25e-06</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 4.881</td> <th>  Cond. No.          </th> <td>3.35e+04</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 3.35e+04. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                    GPA   R-squared:                       0.565\n",
       "Model:                            OLS   Adj. R-squared:                  0.555\n",
       "Method:                 Least Squares   F-statistic:                     52.70\n",
       "Date:                Mon, 11 Jan 2021   Prob (F-statistic):           2.19e-15\n",
       "Time:                        22:09:50   Log-Likelihood:                 25.798\n",
       "No. Observations:                  84   AIC:                            -45.60\n",
       "Df Residuals:                      81   BIC:                            -38.30\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const          0.6439      0.358      1.797      0.076      -0.069       1.357\n",
       "SAT            0.0014      0.000      7.141      0.000       0.001       0.002\n",
       "Attendance     0.2226      0.041      5.451      0.000       0.141       0.304\n",
       "==============================================================================\n",
       "Omnibus:                       19.560   Durbin-Watson:                   1.009\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):               27.189\n",
       "Skew:                          -1.028   Prob(JB):                     1.25e-06\n",
       "Kurtosis:                       4.881   Cond. No.                     3.35e+04\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 3.35e+04. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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9Es0ha9LIyMjAoUOHkJiYiOrqagQHByMwMBDdujV8+EthYSGWL18ueTxKHFZK0hBz1mxskgi/mYhnx/Rs9gGxqWGxxpzVm7vD2JTfh5RNWsZuW3WlCG17dhW9/aJjv6PGr0OzYrRFsiaNAQMGIDY2Fk5OTrh8+TI0Gg3c3d0blYuOjsaLL76IVatWSRqPEoeVmqo5TS/2QNfB77pDLnZiLj7/SNw2hpSsRKuau7R/V8I8TTXhgd31zgll7FWvOYeemvL7kLJJS8y2jemX2BAwFQkPjtf+vfF/CcNeO//1kb15ytnZGTExMdi4cSNGjx4NX1/fButjY2Nx7733om/fviZtv6lRALcbN9QTXp4tEJt0AoVXy9C2tRumjOmFof07mrRvuew5+hdik3NQUaUBUHugiU3OgZdnC6t7L1JRt3ZDwdUy7PQab9TrhEW1I/HGzd0BXWPyrhRX1LZvG2nP0b8afO9cnR1QUdX4Xgp1azeTtt9carWnSb+PK3qSnKn1ZGjb36weX/uPxeK2Mf3NFBRcLWu0vK6ejfktyfG5yEExQ27LysoQGRmJ4OBgTJw4EQBw8uRJLFmyBJs2bUJeXh6mTJlidJ+GsUNubYG5hloqVXPO/MzVL6Gvjj3cnODq7GhUbLomEXRUASoHFao1t7672plMYdl7FJozLFnK7+KrH+1Hn0PJeCV5jejXfLMrCwP73jrY66r7unoe1Lu96PhtaXi7oofc5ubmorKyEr169YKbmxuCgoKQk3NrOoXk5GQUFBQgIiICVVVVyM/Px+TJk/Gf//xHxqiVz5Y79I1tx56X/jI2ZW0Qvf3Ls69DJWLeH11NNU6OKpSVV6OkrFpUbHV0NbNoBKClswO8Wzo1SA4ArGrYq9mbfCsqoO6oBgBsElF8ecg/sa/HEO3fPvvPNUgahvp9bPm3ZCpZk8b58+cRExODLVu2AABSU1MRERGhXR8VFYWoqCht2SlTpjBhiGDLHfqG2rGvlV/FPRvF34SVMDEBD/s8anQcug425ZXVKC3X6I1NH30HoNJyDT54KbDBslc/2m9Vw17N0Rlv1P0SqB3lNH2Z7hYJXXXdVL+PLf+WTCVr0ggMDERmZibGjx8PR0dHBAUFISQkBDNmzEBUVBT8/f3lDM9q2VKH/u10/YDr+iXEdmDXb3JqTrPC7QcbYw5U9RlzYLLGM19jO+Pb3tkGqupq0eV1DYU118Heln9LppK9I3zOnDmYM2dOg2Xr169vVK5Dhw6S3qNhS+p+oAn7/rS50VM+Xq7YppmFG47nRL/GHPM4iWHqgcqYA5Mtnvk6/5iOVhFjRZcvOn4CNXf6NVnGXAd7JcxzpTSyJw2SxqDe7TFu6N020Tm3/8KPCNsRcmuBgaeQnp5xER7O4kfNmYupBypjDky2cuZrTJNTxegQFMduabDM0GAIcx7s5XzgkRIxaZDi1Ag1aP9xK9Hl/959Id4Z9X/SBSRScw5UYg9M1nrma0q/hD5iB0PwYC8NJg1SBDmmDteluTdyWeJAZQ0Hw9aDHoBT7h+iyxszRYetzoFlLZg0SBZz90Thi983iS5viX4Je5vF1ZwczvwJnwHib8C9/sVXqBw1xqR9WeNgAFvCpEEWceb6nxjwpfiDym9T/0A793YSRtSYnGew1jhVhTmbnIxhi4MBrAmTBknGmCanF/u9hNcHLZEwGsPkOoO1liscuZLE7WxlMIC1YtIgs1FKv4Sp5DqDVWobvcf8uXDbWDv8XS2ifMG5fKBFC2mDgvUOBrAVTBpksi0nNuMfPzwvurzYKTr0kboJR64z2KaucKYvS2vWezWqzm7ehLqL+H3cjHoFpdGLjY7JHKxhMICtYtIg0UqrStF1/R2iy38/YQ/ub/eAyfurf8Br2cIRFVU12gn89DXh6DpIAuLOSsWewd6+jz7dfZCZW2RyMmvq+RlNvVdDxDR7yd3kZI19OeZire9dMbPcSsUeZ7mtY46ZN41pchp85yPYPv7bZu2vjq7ZR3WpP9uosbPFDurd3ug6EhNX/e2ba5uA8TPD6pqhVTt1uEgF+cWSzeBqaIZZa2KO75FS3ruiZ7kl5Ync9Ry2nYoXXV6qfgld7fy61D8o6pstFpqGJw3N6S8QE5ex27/9CkcfYzvki4or8GhWGl5OiRH/msPHUdO1m+GCZqDUvhxLsOb3zqRh5zILjmNEfIDo8n/NKoCro/RDG8UeIOt3UhtzUDV1RJTY1zXnCXtNPcPBIEGA2tcbAPCNiP1q/DrgyjHdTwmUmj3fb2HN751Jw84YO0VH/NgdCOw4TLqA9DDUzg807qQW85r6ZaWKqznbB4zvkDe2XyLxh5OKOJu15/strPm9M2nYAWP6JR66YxC+CUuRMBpxdB04HVWAWwsnlJRV6+w41PcaXX0apo6I0rWP2zV3xJWhDnljk8TUxUmK7Gy15/strPm9M2nYoPic/+KF1JmiyyvtfgnAtLH4+l5j7HaMjau5o6f07aduG45Zv6HNsHtEv7Z43WeoGH/rYWbvNisS6ch1v4USRi1Z870mHD1lAwrLCnHvZ+I7L3Omn0HrFm0kjMh6KPXZznIPha1PqXVkCqlGLdlSHXH0lI0ypskp5tGPMannUxJGQ82lpCRhy6x51JJSMGlYiVnfT8P2P7aKKjvozsHYMT7Jps5+bI1n5HNosU380OaCs5cBNzcJI7IP1jxqSSnMnjR+/PFHxMfHIyZG/Nhwauzgxf14LEH81NHNnaKDJFZSAnW3O0UXL3vyaZSsEfnQcxLNmkctKYVZksalS5ewdetWbNu2DZcuXTLqtWvWrEFKSgpUKhUmTJiAadOmNVi/e/dufPDBBxAEAR06dMDSpUvh7e1tjrAtylDnW4WmAh0/FTMtXK1hNz5Ga0c/RdxBSrqxyUl5rHnUklKYnDSqq6uRmpqK+Ph4HDx4EDU1NRAEAV26dEF4eLiobWRkZODQoUNITExEdXU1goODERgYiG7dajt1S0pKsHjxYmzduhW+vr5Ys2YNPvjgA0RHR5satiz0zQH0fvY8/HA5QdQ2/oZZ8C1ueOXBtlhlYZJQPmsetaQURieN06dPIz4+Hjt27MDVq1cBAG5ubggODkZ4eDgeeED8BHUDBgxAbGwsnJyccPnyZWg0Gri7u2vXV1VVYdGiRfD19QUA9OjRA998I+Y+V2Wp63y76piD/S3rPcv6sv7XeLp4Iffv57V/T1+WprOcFG2xlh6SaM79WTJ21+1fw2vWdNHlrxz6GZpud0kSC4nHGXKbR1TSKC8vR1JSEuLj43Hs2DEIggBHR0c8/PDD2L9/P8aNG4fFixebFICzszNiYmKwceNGjB49WpsgAKB169YYOXKkNoZ169bhmWeeMWr7TQ0dk1p5dTk+zPgQnwvzAC/D8yhVvVYFJwfdH4m6tRsKrpbpXK5We+rdZlPrdNlz9C/EJuegokoDoDYpxSbnwMuzBYb272jUtiy9P1O3JbqOBAFwcBAfUNu2QEGB9k9rHuRs7PfIHtlLHTWZNH777TfEx8fj22+/RUlJCQCgb9++CA0NRXBwMHx8fNCzZ89mBxEVFYUZM2YgMjIScXFxmDhxYoP1N27cwAsvvICePXsiLCzMqG1b8j4NQRCQciYJbx58Haeunby1Qk//dAQ24ePna5vyDmbl4e9v7tZ7hjx+SFedbbHjh3TVO0LKlNFTm3ZmaQ+6dSqqNNi0Mwu9O7Uyalv61L8acFABt388pu7PlNgN1VGzm5xu27ZUV0JSXmFxFJ5htlRHzbpPY8KECXBwcMB9992HkSNHYsyYMejQoYPZgsvNzUVlZSV69eoFNzc3BAUFIScnp0GZ/Px8PPfccxg4cCD+9a9/mW3f5nKi6He8c/gNpJxJarJcp+rhuLtsEtyE2s5uFycHTBlTm3DFPPfAUm2xUg9JvP296svnpuzPHLG3Ch4B5yMZossb0y8h1WNdreVxsWQbDDZPubi4oHXr1nBxcUFFhXnbz8+fP4+YmBhs2bIFAJCamoqIiFvTH2g0GkRGRmLMmDF4/nnxT4iT0pXyIrx/dBU++WVtk+UGtB+I6EFvYOAdgwA0fSYo9oYjS7TFSj0kUeyU56bsz6TYz5yBumtX0fu4vvkrVAaJHwpdn1Q3lvGGNbKkJpNGXFwcEhIS8N133yE9PR0qlQp33XUXxo4di5CQEPj5+TVr54GBgcjMzMT48ePh6OiIoKAghISEYMaMGYiKikJeXh5+//13aDQapKTUTqJ333334e23327Wfo1RpanCf7K/wJKDr+NGpf6zyrZuarw+aAkev2cSHB0cG61v6oCvpBuOpB6SKOY9mbo/sbHLNcpJqs9ZSd8fsn1NJo0+ffqgT58+WLBgAfbs2YOEhATs3bsXq1evxnvvvYe+fftCpVKhOdNXzZkzB3PmzGmwbP362ofZ+/v7Izs72+Rtm6q0qhSPJz6GI5ebbqb4xwNz8WK/f8DbtVWz9qekG46kbgbT917r+jaasz99sY8zYrI/QLqhsFJ9zkr6/pDtM3rCwuvXr2Pnzp3YsWMHMjMzAQCOjo4YOHAgxo4di5EjR6Jly5aSBGsKUzrC/7nnJcT+vrHR8rHdx2PBgNdwV+u7zRUeAPuaRM0Sj7l0+/RDeLy2QHT5ggtFgLOzWfbdFKneu9R1qsTvkdLYUh0Z6ghv1iy3Z86cwfbt2/HNN9/g4sWLUKlUcHV1xfDhw7Fq1SpTN2tWpiSNgxf3Y/K3j6OzVxe8PmgJhnUcLvkUHVKMflHqF9ns77W8HOpO7UQXv7HiPZRPfQ6A5euIo6dsky3VkdmSRmVlJYqLi9GqVSs4OTVu1crIyMD27dvx/fff4+bNmzhx4oTpUZuRPUyNro8tfZFvPyhuWmxcZ7S+JidbqiOpsI4Ms6U6avbU6NnZ2VixYgUOHz6MmpoauLi4YNiwYZg3bx7uvPPWBGwDBgzAgAEDsHjxYuzevds80ROhNmH4PTcZm04fEf0aTtFBJI0mrzRyc3PxxBNPoLS0FE5OTvD29saVK1cgCALatm2rnRNKyXilYZ1nP4452WjzyADR5QtP/AnBx8fo/VhzHVkK68gwW6ojQ1caTc6J8Omnn6K0tBQvv/wyjhw5gv379+Onn37CM888g8LCQmzc2LizmMgkggB1Oy/tf4YSxvqh0zH2lQQU5BejIL/YpIRBRMZrsnnqyJEjCAwMxKxZs7TLPDw8sHDhQhw/fhz79++XPECyXcbeLzH2lYQGf3NIKZHlNXmlUVhYiB49euhc179/f1y8eFGSoMg2ub+7tMHVhCF1VxGJP5xExLzEBuv4DAQieTR5pVFZWQkXFxed6zw8PFBW1njWVVI+cwzPFLMNVX4+2t4nfirwop8yUdO5S6PlfAYCkXLwGeF2xhyT2zW1DWPuvi5/fBJufLhOVNnbE8e29FyjYiYi82DSsDPmmNyu/ja+XvMEXDWVtStWGH6tqUNhOZMrkTIYTBpS3wlNltXcye1cdqdg0+LHRe+v4OIVQMfNoMbiTK5EymDw17x27VqsXat/GvBevXo1WqZSqfD77783LzKShNGT21VVQe0nfjjrtbgEVA191NTw9OJMrkTK0GTSqH/HN9kGMdOHGzMUNrv9PXh18gqzTzp4O87kSqQMTSaNtLQ0S8VBFqJrJNJrv32F7sM2id5GQX6xpBPk6SL1cz6ISByjGpsrKirg6lp7Zpednd3oWRcqlQqhoaFwdGz8ECJSjsHuNzHOiAn/Ck5fBDwaTitgiacI3r4/gMNuieQmKml8+eWX+Pe//42IiAi8+OKLAIDdu3fjww8/1JYRBAEqlQp5eXkN7iAnBRAEqH29RRe/viEWlWPHN1nG0lcagOUTFRE1ZjBpLFy4ENu2bUPLli113ug3f/58AEBNTQ0++eQTfPLJJ5g0aRK8vcUfpMj8vMNDgX17oRZRVuPbHld+PSl62xz+SmS/mkwa+/fvx9atWzF48GCsWrUKrVq1alTm2Wef1f7b09MT0dHR2Lp1K6ZPn272YEk/l5QkeD8zUXT55kwdbk3DX+W4IjKFtcRJ1GTS+Prrr+Hp6YnVq1eLunIICwvDe++9h71794pOGmvWrEFKSgpUKhUmTJiAadOmNVh/4sQJLFy4EKWlpXjwwQfxxhtv6HwIlL1RXbuKtvd0Fl2+KDMHNe3vMMu+pRr++kVKNtKPX0SNUPvM8MD778Qzo3qavD1ruSKyljiJAAMTFh47dgwBAQGim5ocHR0xZMgQ/PHHH6LKZ2Rk4NChQ0hMTMTWrVvxxRdf4PTp0w3KvPrqq3j99deRkpICQRAQFxcnatu2qP5kf4YSRvEHnwCCoJ30z1wJA9A/zLU5w1+/SMnGD8dqEwYA1AjAD8cu4ouU7KZf2ISmroiUxFriJAIMJI2ioiJ06NBB57oePXogNDS00XJfX19cv35d1M4HDBiA2NhYODk5oaioCBqNBu7u7tr1Fy5cQHl5Oe6//34AQHh4OJKTk0Vt2xZ4LPin6Flhq7vfpU0QBfnFqJg4WbK4wgO7w8Wp4VenucNf04/rnjFZ33IxrOWGQGuJkwgw0Dzl5eWF0tJSnetGjhyJkSNHNlp+7do1tGnTRnQAzs7OiImJwcaNGzF69OgGTwLMz8+HWn2rK1etVuPy5cuitw2gySdQKc7Ro8CDD4ovr9EADrUHbydAZ6e3Wu1pltDqGzfUE16eLRCbdAKFV8vQtrUbpozphaH9O5q8TX0PV6wRTH8P6tZuKLjaeCZmdWu3BtuUoo6MITZOOSklDiWzlzoyeEf4zz//bNQGDx8+jE6dOhn1mqioKMyYMQORkZGIi4vDxIm1Hbo1NTUN5r6qG9ZrDEU/7lWjgfqO1qKLX9l/BJq7680iW6Q7odeR8hGUvTu1wvJZgxosa86+HFS6E4eDyvTtjh/SVecNgeOHdNVuUwmP6RQTp5yUUEdKZ0t11KzHvQ4fPhwnTpzAoUOHRO1s165dOHv2LEaNGiWqfG5uLk6cOAEAcHNzQ1BQEHJycrTr27dvj4KCAu3fhYWFaNeunahtK1XL6P+71eRkIGGU/nN+gyanBgnDxgTer3vKGn3LxRjUuz2eHdNT29fi4+Uq6VQnprKWOIkAA1ca4eHhWL9+PV555RWsWrUKgwYN0lv2yJEjiI6Oho+PD8aNGydq5+fPn0dMTAy2bNkCAEhNTUVERIR2vZ+fH1xdXXH06FH0798fO3bsQEBAgKhtK4WqoAAtVy+H2wbDz42ovqcHru77yQJRKU/dKClzjp4CrOeGQGuJk0glCEKTbTfJycl4+eWXAQBDhw7FyJEjcffdd8Pb2xvXr1/HuXPn8P3332P37t0QBAHr16/H4MGDRQfwwQcfICkpCY6OjggKCsKcOXMwY8YMREVFwd/fH9nZ2YiOjkZJSQl69+6NpUuX6n2aoC4Wb56qrESL2I3wWPI6VOXlBosXXCgCnJ0lCcWWLpmlwjoyjHVkmC3VkaHmKYNJAwAOHDiA6OhoXLx4UWefgiAI8PX1xYoVK/DQQw81L2IzkzxpCAKcf9gNjyWL4PT7b00WrRg7HiXvrECNr2XOKG3piywV1pFhrCPDbKmODCUNUXfJPfzww0hJSUF6ejpSU1Nx7tw5FBUVoVWrVvDz88Pw4cMxfPhw7WSGts7xj1No+c4SuO7c0WS58vDHUTo/GjVdulooMiIiaYm+tdrZ2RkjRozAiBEjpIxHkVTXr8H9g/fhHrO6yXJV/R5AafQbqHok0EKRERFZFufj0EUQ4PrVf+DxRjQcior0FqvxboXS195A+ZNPS9YvQUSkJEwaOrivWo6WK97Rue7m81G4GfUyhDbiH4FKRGQrmDR00PjdmjqlYnQwShe8Dk2ve82+H85sSkTWhklDh4onn0bBk09Lug/ObEpE1qjJO8JJOpzZlIisEZOGTDizKRFZIyYNmUjxTAoiIqkxachEimdSEBFJjR3hMqnr7OboKSKyJkwaMuLMpkRkbdg8RUREojFpEBGRaEwaREQkGvs0rBinISEiS2PSsFKchoSI5MDmKSvFaUiISA5MGlaK05AQkRxkTxpr165FSEgIQkJCsGLFikbrs7KyEBERgXHjxmHWrFkoLi6WIUrl4TQkRCQHWZPGgQMHsG/fPmzfvh0JCQnIysrCrl27GpR5++23ERUVhcTERHTt2hUbNmyQKVpl4TQkRCQHWTvC1Wo15s+fDxcXFwBA9+7dcfHixQZlampqUFpaCgAoKyuDt7e3xeNUIk5DQkRyUAmCIMgdBACcOXMGTz75JLZs2YIuXbpolx8/fhzTp0+Hu7s73NzcEBcXh9atW8sXKBGRHVNE0jh16hRmzZqFOXPmICwsTLu8vLwcERERWLp0Kfr06YPPPvsMBw8exLp160Rvu6ioBDU1sr9FWajVnigouCF3GIrGOjKMdWSYLdWRg4MKPj4e+tdbMBadjh49iqlTp2Lu3LkNEgYAnDx5Eq6urujTpw8AYOLEicjIyJAjTCIigsxJ49KlS3jhhRewcuVKhISENFrfuXNn5OXl4fTp0wCA1NRU+Pv7WzpMIiL6H1k7wjds2ICKigosW7ZMu2zSpElIS0tDVFQU/P39sXTpUrz00ksQBAE+Pj545513ZIyYiMi+KaJPQ0rs07CNdlapsI4MYx0ZZkt1ZKhPg3NPWQlOTkhESsCkYQU4OSERKYXso6fIME5OSERKwaRhBTg5IREpBZOGFeDkhESkFEwaVoCTExKRUrAj3ApwckIiUgomDSsxqHd7Jgkikh2bp4iISDQmDSIiEo1Jg4iIRGPSICIi0Zg0iIhINCYNIiISjUmDiIhEY9IgIiLRmDSIiEg0Jg0iIhKNSYOIiESTfe6ptWvXIikpCQAQGBiIefPmNVh/+vRpLFq0CNevX4darcbq1avh7e0tR6hERHZP1iuNAwcOYN++fdi+fTsSEhKQlZWFXbt2adcLgoDZs2djxowZSExMRK9evbBu3ToZIyYism+yXmmo1WrMnz8fLi4uAIDu3bvj4sWL2vVZWVlwd3dHQEAAACAyMhLFxcWyxEpERIBKEARB7iAA4MyZM3jyySexZcsWdOnSBQDw3XffYfv27VCr1Thx4gS6deuG1157Da1atZI1ViIieyV7nwYAnDp1CrNmzcK8efO0CQMAqqurkZGRgc2bN8Pf3x/vv/8+li1bhmXLlonedlFRCWpqFJEXLU6t9kRBwQ25w1A01pFhrCPDbKmOHBxU8PHx0L/egrHodPToUUydOhVz585FWFhYg3VqtRqdO3eGv78/ACA0NBSZmZlyhElERJA5aVy6dAkvvPACVq5ciZCQkEbr+/XrhytXriA7OxsAkJaWht69e1s6TCIi+h9Zm6c2bNiAioqKBs1NkyZNQlpaGqKiouDv748PP/wQ0dHRKCsrQ/v27bFixQoZIyYism+K6QiXCvs0bKOdVSqsI8NYR4bZUh0Z6tNQREc42a+DWXnYlp6LouIK+Hi5IjywOwb1bi93WESkB5MGyeZgVh4+T8pGZXUNAKCouAKfJ9X2XzFxECmT7KOnyH5tS8/VJow6ldU12JaeK1NERGQIrzQshM0wjRUVVxi1nIjkxysNC6hrhqk7GNY1wxzMypM5Mnn5eLkatZyI5MekYQFshtEtPLA7XJwafgVdnBwQHthdpoiIyBA2T1kAm2F0q2ueY7MdkfVg0rAAHy9XnQmCzTC1iYNJgsh6sHnKAtgMQ0S2glcaFsBmGCKyFUwaFsJmGCKyBWyeIiIi0Zg0iIhINCYNIiISjUmDiIhEY9IgIiLRmDSIiEg0Jg0iIhKNSYOIiEST/ea+tWvXIikpCQAQGBiIefPm6Sy3Z88eLFmyBGlpaZYMz6bxGR9EZCxZrzQOHDiAffv2Yfv27UhISEBWVhZ27drVqFxhYSGWL18uQ4S2i8/4ICJTyJo01Go15s+fDxcXFzg7O6N79+64ePFio3LR0dF48cUXZYjQdvEZH0RkClmbp+6++27tv8+cOYOkpCRs2bKlQZnY2Fjce++96Nu3r0n78PHxaFaM1k6t9tS5/IqeZ3lcKa7Q+xpbZW/v1xSsI8PspY5k79MAgFOnTmHWrFmYN28eunTpol1+8uRJfP/999i0aRPy8kxrNikqKkFNjWCmSK2LWu2JgoIbOte10fOMjzZernpfY4uaqiOqxToyzJbqyMFB1eTJtuyjp44ePYqpU6di7ty5CAsLa7AuOTkZBQUFiIiIwMyZM5Gfn4/JkyfLFKlt4TM+iMgUKkEQZDsNv3TpEsLCwvDee+9h0KBBTZY9f/48pkyZYvToKV5p6D/74egp2zpDlArryDBbqiNDVxqyNk9t2LABFRUVWLZsmXbZpEmTkJaWhqioKPj7+8sYne3jMz6IyFiyXmlYAq80bOPsRyqsI8NYR4bZUh0pvk+DiIisB5MGERGJxqRBRESiKeI+DSk5OKjkDkFW9v7+xWAdGcY6MsxW6sjQ+7D5jnAiIjIfNk8REZFoTBpERCQakwYREYnGpEFERKIxaRARkWhMGkREJBqTBhERicakQUREojFpEBGRaEwaVqikpAShoaE4f/480tPT8dhjj2n/GzhwIGbNmgUAOHHiBMLDwzFq1CgsXLgQ1dXVAICLFy/iqaeewujRozF79myUlpbK+XYkUb+OAGDfvn0YN24cQkNDMW/ePFRWVgJgHdWvo23btiE4OBhjx47FW2+9pa0Le62jtWvXIiQkBCEhIVixYgUA4MCBAxg7diyCgoLw3nvvacvaVR0JZFWOHz8uhIaGCr179xb++uuvBuvy8/OF4cOHC3/++acgCIIQEhIiHDt2TBAEQViwYIHw5ZdfCoIgCDNnzhR27twpCIIgrF27VlixYoXF4rcEXXUUEBAg/PHHH4IgCMKcOXOEuLg4QRBYR3V1lJubKzzyyCPC5cuXBUEQhEWLFgkbN24UBME+62j//v3CxIkThYqKCqGyslKYMmWK8M033wiBgYHCuXPnhKqqKmH69OnCnj17BEGwrzrilYaViYuLw6JFi9CuXbtG61asWIFJkyahS5cuuHDhAsrLy3H//fcDAMLDw5GcnIyqqir89NNPGDVqVIPltkRXHWk0GpSUlECj0aCiogKurq6so3p1lJOTg/vvv1/797Bhw7B79267rSO1Wo358+fDxcUFzs7O6N69O86cOYPOnTujY8eOcHJywtixY5GcnGx3dWTzs9zamrffflvn8jNnziAjI0O7Pj8/H2q1WrterVbj8uXLuHr1Kjw8PODk5NRguS3RVUeLFy/GM888Aw8PD3To0AGjR49GVlYW6+h/evbsiWXLluHSpUto164dkpOTUVhYaLffo7vvvlv77zNnziApKQlPP/10g7po164dLl++bHd1xCsNG/HVV19h8uTJcHFxAQDU1NRApbo1xbEgCFCpVNr/13f737amoKAAK1euxM6dO7Fv3z707dsXS5cuZR3V07VrV8ydOxezZ8/GU089hR49esDZ2dnu6+jUqVOYPn065s2bh44dO+qsC3urIyYNG5Gamorg4GDt3+3bt0dBQYH278LCQrRr1w5t2rTBjRs3oNFoANQeUHU1ddmSI0eO4J577kGnTp3g4OCAJ554AhkZGayjeioqKtCnTx8kJCTgv//9L3x9fdGxY0e7rqOjR49i6tSpmDt3LsLCwhrVRd17trc6YtKwAVeuXEF5eTk6duyoXebn5wdXV1ccPXoUALBjxw4EBATA2dkZDz74IL777jsAQEJCAgICAmSJ21LuueceZGZmorCwEEBtgvX392cd1XPz5k1MnToVJSUlqKysxObNmxEcHGy3dXTp0iW88MILWLlyJUJCQgAAffv2xZ9//omzZ89Co9Fg586dCAgIsLs64kOYrNSjjz6K2NhYdOjQAZmZmXjrrbcQFxfXoEx2djaio6NRUlKC3r17Y+nSpXBxccGFCxcwf/58FBUV4Y477sDq1avh7e0t0zuRTv062r59O9avXw9HR0d07twZS5YsQZs2bVhH9eooPj4emzZtQnV1NUJDQzFnzhwA9vk9euutt7B161Z06tRJu6xukMnSpUtRUVGBwMBALFiwACqVyq7qiEmDiIhEY/MUERGJxqRBRESiMWkQEZFoTBpERCQakwYREYnGaUSImiE1NRVxcXHIzMzEjRs30KpVK/j7+2PChAkYPny43tetW7cOq1atQqtWrfDjjz9q7+QHamebXbBggegYcnJymvUeiIzBpEFkojfffBObN2+Gn58fhg8fjtatW+Py5ctIT09HWloannjiCbz55ps6X5uYmAg3Nzdcu3YN33//PUJDQ7XrevXqhRdffLFB+d27dyM7OxthYWHw8/OT9H0RNYVJg8gEhw8fxubNmzFq1CisXr1aOykdANy4cQNTpkxBXFwcAgMDMWLEiAav/e2333Dq1ClERkZiw4YNiI+Pb5Q0evXq1eA1Fy5c0CaNhx56SNo3R9QE9mkQmWDPnj0AgKeeeqpBwgAAT09PzJ07FwCwa9euRq9NSEgAAIwaNQoDBw7E4cOH8ddff0kaL5G5MGkQmaCqqgoAcPLkSZ3rH3zwQbz//vuYOnVqg+XV1dX47rvv0LZtW/Tq1QvBwcEQBAFff/211CETmQWTBpEJBg8eDABYvnw53nzzTRw7dkw7mykAtGjRAmPGjGnUzLR3714UFRVh9OjRUKlUGDlyJFxcXLBt27YGrydSKiYNIhMMGzYMTz75JKqqqrB582ZMmjQJAwYMwMyZM7Fp0ybk5eXpfF1d01TdzKmenp4IDAxEfn4+0tPTLRU+kcmYNIhMtHjxYnz66ad45JFH4OzsjJKSEqSnp2Pp0qUYMWIEVq1ahZqaGm354uJi/PDDD/Dz80O/fv20y+s6wePj4y3+HoiMxdFTRM0wdOhQDB06FKWlpThy5AgOHjyItLQ0nD17FuvWrUNNTQ1effVVAEBSUhIqKysRHBzc4Aluw4YNg4eHB/bu3Yv8/HybeFAP2S5eaRCZQcuWLREYGIj58+cjJSUFb731FlQqFTZv3oyysjIAt5qm1q9fjx49emj/69OnD0pKSlBdXY3t27fL+C6IDOOVBpGRSkpKEB4ejq5du+LTTz9ttF6lUuHxxx9HcnIy9u3bh7y8PDg5OeHnn3+Gr68vhg4d2ug1paWl2LlzJ77++mvMnDnTJp4lTbaJSYPISB4eHrhx4wYOHDiAwsJCtG3bVm9ZBwcHqNVqfPbZZwBqn/72/PPP6yz766+/4uzZszh8+DAGDhwoSexEzcXmKSITPPXUU6isrERUVBTy8/MbrU9NTcWBAwcwcuRIeHh4IDExEQAwduxYvdsMCwsDwA5xUjZeaRCZYPbs2Th58iRSUlIQFBSEIUOGoEuXLqiursYvv/yCn3/+Gd26dcPixYtx5MgRnDt3Dv369UPHjh31bjMsLAwxMTHYtWsXrl+/bvXPkibbxCsNIhM4OjoiJiYGa9euxSOPPIJff/0VsbGxiI+PR0VFBebOnYvt27ejTZs22quMcePGNbnN9u3b4+GHH0ZFRYX2NURKoxIEQZA7CCIisg680iAiItGYNIiISDQmDSIiEo1Jg4iIRGPSICIi0Zg0iIhINCYNIiISjUmDiIhEY9IgIiLRmDSIiEi0/wdMNfX6fJSf7gAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(data2[\"SAT\"],y)\n",
    "yhat_no = 0.6439 + 0.0014 *data2[\"SAT\"]\n",
    "yhat_yes = 0.8665 + 0.0014*data2[\"SAT\"]\n",
    "fig = plt.plot(data2[\"SAT\"], yhat_no, lw=2, c=\"red\")\n",
    "fig = plt.plot(data2[\"SAT\"], yhat_yes, lw=2, c=\"green\")\n",
    "plt.xlabel(\"SAT\", fontsize=20)\n",
    "plt.ylabel(\"GPA\", fontsize=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyError",
     "evalue": "'Attendance'",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\pandas\\core\\indexes\\base.py\u001b[0m in \u001b[0;36mget_loc\u001b[1;34m(self, key, method, tolerance)\u001b[0m\n\u001b[0;32m   2894\u001b[0m             \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2895\u001b[1;33m                 \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_engine\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcasted_key\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2896\u001b[0m             \u001b[1;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m \u001b[1;32mas\u001b[0m \u001b[0merr\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32mpandas\\_libs\\index.pyx\u001b[0m in \u001b[0;36mpandas._libs.index.IndexEngine.get_loc\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;32mpandas\\_libs\\index.pyx\u001b[0m in \u001b[0;36mpandas._libs.index.IndexEngine.get_loc\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;32mpandas\\_libs\\hashtable_class_helper.pxi\u001b[0m in \u001b[0;36mpandas._libs.hashtable.PyObjectHashTable.get_item\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;32mpandas\\_libs\\hashtable_class_helper.pxi\u001b[0m in \u001b[0;36mpandas._libs.hashtable.PyObjectHashTable.get_item\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;31mKeyError\u001b[0m: 'Attendance'",
      "\nThe above exception was the direct cause of the following exception:\n",
      "\u001b[1;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-43-1dde383b4e96>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mscatter\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdata2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"SAT\"\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0my\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mc\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mdata\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"Attendance\"\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcmap\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m\"RdYlGn_r\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      2\u001b[0m \u001b[0myhat_no\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0.6439\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m0.0014\u001b[0m \u001b[1;33m*\u001b[0m\u001b[0mdata2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"SAT\"\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      3\u001b[0m \u001b[0myhat_yes\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0.8665\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m0.0014\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mdata2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"SAT\"\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[0myhat\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0.0017\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mdata\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"SAT\"\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m0.275\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0mfig\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdata2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"SAT\"\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0myhat_no\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mlw\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mc\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m\"red\"\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mlabel\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m\"regression line1\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\pandas\\core\\frame.py\u001b[0m in \u001b[0;36m__getitem__\u001b[1;34m(self, key)\u001b[0m\n\u001b[0;32m   2900\u001b[0m             \u001b[1;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcolumns\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mnlevels\u001b[0m \u001b[1;33m>\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2901\u001b[0m                 \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_getitem_multilevel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2902\u001b[1;33m             \u001b[0mindexer\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcolumns\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2903\u001b[0m             \u001b[1;32mif\u001b[0m \u001b[0mis_integer\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mindexer\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2904\u001b[0m                 \u001b[0mindexer\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m[\u001b[0m\u001b[0mindexer\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\pandas\\core\\indexes\\base.py\u001b[0m in \u001b[0;36mget_loc\u001b[1;34m(self, key, method, tolerance)\u001b[0m\n\u001b[0;32m   2895\u001b[0m                 \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_engine\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcasted_key\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2896\u001b[0m             \u001b[1;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m \u001b[1;32mas\u001b[0m \u001b[0merr\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2897\u001b[1;33m                 \u001b[1;32mraise\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mfrom\u001b[0m \u001b[0merr\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2898\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2899\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mtolerance\u001b[0m \u001b[1;32mis\u001b[0m \u001b[1;32mnot\u001b[0m \u001b[1;32mNone\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mKeyError\u001b[0m: 'Attendance'"
     ]
    }
   ],
   "source": [
    "plt.scatter(data2[\"SAT\"],y,c=data[\"Attendance\"], cmap=\"RdYlGn_r\")\n",
    "yhat_no = 0.6439 + 0.0014 *data2[\"SAT\"]\n",
    "yhat_yes = 0.8665 + 0.0014*data2[\"SAT\"]\n",
    "yhat = 0.0017*data[\"SAT\"] + 0.275\n",
    "fig = plt.plot(data2[\"SAT\"], yhat_no, lw=2, c=\"red\", label=\"regression line1\")\n",
    "fig = plt.plot(data2[\"SAT\"], yhat_yes, lw=2, c=\"green\", label=\"regression line2\")\n",
    "fig = plt.plot(data2[\"SAT\"], yhat, lw=2, c=\"yellow\", label=\"regression line\")\n",
    "plt.xlabel(\"SAT\", fontsize=20)\n",
    "plt.ylabel(\"GPA\", fontsize=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyError",
     "evalue": "'Attendance'",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\pandas\\core\\indexes\\base.py\u001b[0m in \u001b[0;36mget_loc\u001b[1;34m(self, key, method, tolerance)\u001b[0m\n\u001b[0;32m   2894\u001b[0m             \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2895\u001b[1;33m                 \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_engine\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcasted_key\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2896\u001b[0m             \u001b[1;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m \u001b[1;32mas\u001b[0m \u001b[0merr\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32mpandas\\_libs\\index.pyx\u001b[0m in \u001b[0;36mpandas._libs.index.IndexEngine.get_loc\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;32mpandas\\_libs\\index.pyx\u001b[0m in \u001b[0;36mpandas._libs.index.IndexEngine.get_loc\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;32mpandas\\_libs\\hashtable_class_helper.pxi\u001b[0m in \u001b[0;36mpandas._libs.hashtable.PyObjectHashTable.get_item\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;32mpandas\\_libs\\hashtable_class_helper.pxi\u001b[0m in \u001b[0;36mpandas._libs.hashtable.PyObjectHashTable.get_item\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;31mKeyError\u001b[0m: 'Attendance'",
      "\nThe above exception was the direct cause of the following exception:\n",
      "\u001b[1;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-43-1dde383b4e96>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mscatter\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdata2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"SAT\"\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0my\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mc\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mdata\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"Attendance\"\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcmap\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m\"RdYlGn_r\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      2\u001b[0m \u001b[0myhat_no\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0.6439\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m0.0014\u001b[0m \u001b[1;33m*\u001b[0m\u001b[0mdata2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"SAT\"\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      3\u001b[0m \u001b[0myhat_yes\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0.8665\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m0.0014\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mdata2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"SAT\"\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[0myhat\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0.0017\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mdata\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"SAT\"\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m0.275\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0mfig\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdata2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;34m\"SAT\"\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0myhat_no\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mlw\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mc\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m\"red\"\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mlabel\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m\"regression line1\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\pandas\\core\\frame.py\u001b[0m in \u001b[0;36m__getitem__\u001b[1;34m(self, key)\u001b[0m\n\u001b[0;32m   2900\u001b[0m             \u001b[1;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcolumns\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mnlevels\u001b[0m \u001b[1;33m>\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2901\u001b[0m                 \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_getitem_multilevel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2902\u001b[1;33m             \u001b[0mindexer\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcolumns\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2903\u001b[0m             \u001b[1;32mif\u001b[0m \u001b[0mis_integer\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mindexer\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2904\u001b[0m                 \u001b[0mindexer\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m[\u001b[0m\u001b[0mindexer\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\pandas\\core\\indexes\\base.py\u001b[0m in \u001b[0;36mget_loc\u001b[1;34m(self, key, method, tolerance)\u001b[0m\n\u001b[0;32m   2895\u001b[0m                 \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_engine\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcasted_key\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2896\u001b[0m             \u001b[1;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m \u001b[1;32mas\u001b[0m \u001b[0merr\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2897\u001b[1;33m                 \u001b[1;32mraise\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mfrom\u001b[0m \u001b[0merr\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2898\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2899\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mtolerance\u001b[0m \u001b[1;32mis\u001b[0m \u001b[1;32mnot\u001b[0m \u001b[1;32mNone\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mKeyError\u001b[0m: 'Attendance'"
     ]
    }
   ],
   "source": [
    "%tb"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# How to make predictions based on the created regressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>const</th>\n",
       "      <th>SAT</th>\n",
       "      <th>Attendance</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1714</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1664</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1760</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1685</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1693</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>79</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1936</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1810</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>81</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1987</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>82</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1962</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>83</th>\n",
       "      <td>1.0</td>\n",
       "      <td>2050</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>84 rows × 3 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "    const   SAT  Attendance\n",
       "0     1.0  1714           0\n",
       "1     1.0  1664           0\n",
       "2     1.0  1760           0\n",
       "3     1.0  1685           0\n",
       "4     1.0  1693           0\n",
       "..    ...   ...         ...\n",
       "79    1.0  1936           1\n",
       "80    1.0  1810           1\n",
       "81    1.0  1987           0\n",
       "82    1.0  1962           1\n",
       "83    1.0  2050           1\n",
       "\n",
       "[84 rows x 3 columns]"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<style scoped>\n",
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       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>const</th>\n",
       "      <th>SAT</th>\n",
       "      <th>Attendance</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>1700</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>1670</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   const   SAT  Attendance\n",
       "0      1  1700           0\n",
       "1      1  1670           1"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_data = pd.DataFrame({\"const\":1, \"SAT\":[1700, 1670], \"Attendance\":[0,1]})\n",
    "new_data = new_data[[\"const\", \"SAT\", \"Attendance\"]]\n",
    "new_data.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>const</th>\n",
       "      <th>SAT</th>\n",
       "      <th>Attendance</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Alice</th>\n",
       "      <td>1</td>\n",
       "      <td>1700</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bob</th>\n",
       "      <td>1</td>\n",
       "      <td>1670</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       const   SAT  Attendance\n",
       "Alice      1  1700           0\n",
       "Bob        1  1670           1"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_data.rename(index={0:\"Alice\", 1:\"Bob\"})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    3.023513\n",
       "1    3.204163\n",
       "dtype: float64"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predictions = results.predict(new_data)\n",
    "predictions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>const</th>\n",
       "      <th>SAT</th>\n",
       "      <th>Attendance</th>\n",
       "      <th>Predictions</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Alice</th>\n",
       "      <td>1</td>\n",
       "      <td>1700</td>\n",
       "      <td>0</td>\n",
       "      <td>3.023513</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bob</th>\n",
       "      <td>1</td>\n",
       "      <td>1670</td>\n",
       "      <td>1</td>\n",
       "      <td>3.204163</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       const   SAT  Attendance  Predictions\n",
       "Alice      1  1700           0     3.023513\n",
       "Bob        1  1670           1     3.204163"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predictionsdf = pd.DataFrame({\"Predictions\": predictions})\n",
    "joined = new_data.join(predictionsdf)\n",
    "joined.rename(index={0:\"Alice\", 1:\"Bob\"})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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