Ayudantía-n°4-Test-de-hipótesis.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "WX_1evN-AHPA"
   },
   "source": [
    "![logo uai 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)\n",
    "\n",
    "\n",
    "**Fundamentos de ciencias de datos,   Sección 3,   2022-2.**\n",
    "\n",
    "\n",
    "**Profesora:   Carolina Lagos Campos.**\n",
    "\n",
    "\n",
    "**Ayudante:   Bruno Reyes Sánchez.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "MxkoPKUT8HXu"
   },
   "source": [
    "## **Tópicos**\n",
    "\n",
    "  - Intervalos de confianza y error\n",
    "  - Test de hipótesis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###### **Resumen métricas estadísticas**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "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>Nombre</th>\n",
       "      <th>Sexo</th>\n",
       "      <th>Edad</th>\n",
       "      <th>Puntaje</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Anita</td>\n",
       "      <td>M</td>\n",
       "      <td>17</td>\n",
       "      <td>630</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Bastian</td>\n",
       "      <td>H</td>\n",
       "      <td>19</td>\n",
       "      <td>450</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Carlos</td>\n",
       "      <td>H</td>\n",
       "      <td>17</td>\n",
       "      <td>550</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Andrea</td>\n",
       "      <td>M</td>\n",
       "      <td>17</td>\n",
       "      <td>680</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Ignacia</td>\n",
       "      <td>M</td>\n",
       "      <td>20</td>\n",
       "      <td>710</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>Antonio</td>\n",
       "      <td>H</td>\n",
       "      <td>17</td>\n",
       "      <td>450</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>Nicholas</td>\n",
       "      <td>H</td>\n",
       "      <td>21</td>\n",
       "      <td>600</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>Vicente</td>\n",
       "      <td>H</td>\n",
       "      <td>19</td>\n",
       "      <td>630</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>Diego</td>\n",
       "      <td>H</td>\n",
       "      <td>20</td>\n",
       "      <td>700</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>Francisca</td>\n",
       "      <td>M</td>\n",
       "      <td>20</td>\n",
       "      <td>550</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>Antonia</td>\n",
       "      <td>M</td>\n",
       "      <td>20</td>\n",
       "      <td>650</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>Catalina</td>\n",
       "      <td>M</td>\n",
       "      <td>21</td>\n",
       "      <td>310</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>Maria</td>\n",
       "      <td>M</td>\n",
       "      <td>18</td>\n",
       "      <td>300</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>Carolina</td>\n",
       "      <td>M</td>\n",
       "      <td>20</td>\n",
       "      <td>480</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>Matias</td>\n",
       "      <td>H</td>\n",
       "      <td>18</td>\n",
       "      <td>780</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>Joaquin</td>\n",
       "      <td>H</td>\n",
       "      <td>21</td>\n",
       "      <td>850</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       Nombre Sexo  Edad  Puntaje\n",
       "0       Anita    M    17      630\n",
       "1     Bastian    H    19      450\n",
       "2      Carlos    H    17      550\n",
       "3      Andrea    M    17      680\n",
       "4     Ignacia    M    20      710\n",
       "5     Antonio    H    17      450\n",
       "6    Nicholas    H    21      600\n",
       "7     Vicente    H    19      630\n",
       "8       Diego    H    20      700\n",
       "9   Francisca    M    20      550\n",
       "10    Antonia    M    20      650\n",
       "11   Catalina    M    21      310\n",
       "12      Maria    M    18      300\n",
       "13   Carolina    M    20      480\n",
       "14     Matias    H    18      780\n",
       "15    Joaquin    H    21      850"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Puntajes PSU\n",
    "\n",
    "df_punt = pd.DataFrame({'Nombre': ['Anita', 'Bastian', 'Carlos', 'Andrea','Ignacia',\"Antonio\", \"Nicholas\", \"Vicente\", \"Diego\", \"Francisca\", \"Antonia\", \"Catalina\", \"Maria\", \"Carolina\", \"Matias\", \"Joaquin\"],\n",
    "                 'Sexo':          ['M','H', 'H','M','M','H','H','H','H',\"M\",\"M\",\"M\",\"M\",\"M\",\"H\",\"H\"],\n",
    "                 'Edad':          [17, 19,17,17,20,17,21,19,20,20,20,21,18,20,18,21],\n",
    "                 'Puntaje':       [630, 450, 550, 680, 710,450,600,630,700,550,650,310,300,480,780,850]})\n",
    "df_punt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "19.0625"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Media poblacional\n",
    "\n",
    "Edad = df_punt['Edad']  #<== Selección de una columna en particular NUMERICA\n",
    "Edad.mean() #<== Calculo de la media/promedio de la población"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.526160760426852"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Desviación estándar\n",
    "\n",
    "Edad.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "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>Edad</th>\n",
       "      <th>Puntaje</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>17</td>\n",
       "      <td>630</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>36</td>\n",
       "      <td>1080</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>53</td>\n",
       "      <td>1630</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>70</td>\n",
       "      <td>2310</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>90</td>\n",
       "      <td>3020</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>107</td>\n",
       "      <td>3470</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>128</td>\n",
       "      <td>4070</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>147</td>\n",
       "      <td>4700</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>167</td>\n",
       "      <td>5400</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>187</td>\n",
       "      <td>5950</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>207</td>\n",
       "      <td>6600</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>228</td>\n",
       "      <td>6910</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>246</td>\n",
       "      <td>7210</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>266</td>\n",
       "      <td>7690</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>284</td>\n",
       "      <td>8470</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>305</td>\n",
       "      <td>9320</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    Edad  Puntaje\n",
       "0     17      630\n",
       "1     36     1080\n",
       "2     53     1630\n",
       "3     70     2310\n",
       "4     90     3020\n",
       "5    107     3470\n",
       "6    128     4070\n",
       "7    147     4700\n",
       "8    167     5400\n",
       "9    187     5950\n",
       "10   207     6600\n",
       "11   228     6910\n",
       "12   246     7210\n",
       "13   266     7690\n",
       "14   284     8470\n",
       "15   305     9320"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#También puede calcular el valor acumulado: (Ignorar el valor NAN)\n",
    "\n",
    "df_punt[['Edad', 'Puntaje']].cumsum()  # <== buscar otro contexto para frecuencia acumulada\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "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>Edad</th>\n",
       "      <th>Puntaje</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>16.000000</td>\n",
       "      <td>16.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>19.062500</td>\n",
       "      <td>582.500000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>1.526161</td>\n",
       "      <td>155.198797</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>17.000000</td>\n",
       "      <td>300.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>17.750000</td>\n",
       "      <td>472.500000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>19.500000</td>\n",
       "      <td>615.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>20.000000</td>\n",
       "      <td>685.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>21.000000</td>\n",
       "      <td>850.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "            Edad     Puntaje\n",
       "count  16.000000   16.000000\n",
       "mean   19.062500  582.500000\n",
       "std     1.526161  155.198797\n",
       "min    17.000000  300.000000\n",
       "25%    17.750000  472.500000\n",
       "50%    19.500000  615.000000\n",
       "75%    20.000000  685.000000\n",
       "max    21.000000  850.000000"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Resumen de metricas estadísticas\n",
    "\n",
    "df_punt.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "# [INSERTAR IMAGEN DE RESUMEN DE MEDIDAS]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "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>Edad</th>\n",
       "      <th>Puntaje</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Edad</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.092179</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Puntaje</th>\n",
       "      <td>0.092179</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             Edad   Puntaje\n",
       "Edad     1.000000  0.092179\n",
       "Puntaje  0.092179  1.000000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "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>Edad</th>\n",
       "      <th>Puntaje</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Edad</th>\n",
       "      <td>2.329167</td>\n",
       "      <td>21.833333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Puntaje</th>\n",
       "      <td>21.833333</td>\n",
       "      <td>24086.666667</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              Edad       Puntaje\n",
       "Edad      2.329167     21.833333\n",
       "Puntaje  21.833333  24086.666667"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Los métodos corr y cov del marco de datos pueden devolver una matriz completa de similitud o varianza:\n",
    "display(df_punt.corr())\n",
    "display(df_punt.cov())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Edad       0.092179\n",
       "Puntaje    1.000000\n",
       "dtype: float64"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Usando el método corrwith de Dataframe, podemos calcular la similitud entre diferentes columnas o filas en el marco de datos. Pasar una serie:\n",
    "\n",
    "df_punt.corrwith(df_punt.Puntaje)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Intervalos de confianza y error "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**¿Cómo cuantificamos el error muestral?:** Podemos usar intervalos de\n",
    "confianza.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se llama intervalo de confianza a un par o varios pares de números entre los cuales se estima que estará cierto valor desconocido respecto de un parámetro poblacional con un determinado nivel de confianza."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos a construir un intervalo de confianza para la **media**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tres casos:\n",
    "- Caso 1: Datos normales y varianza conocida.\n",
    "- Caso 2: Datos normales y varianza desconocida.\n",
    "- Caso 3: Datos de cualquier distribución pero tamaño de muestra grande."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- **Caso 1**: datos normales con varianza conocida.\n",
    "$$a_n=\\bar{X}- z_{1-\\alpha/2}\\frac{\\sigma}{\\sqrt{n}}\\qquad \\text{y}\\qquad b_n=\\bar{X}+ z_{1-\\alpha/2}\\frac{\\sigma}{\\sqrt{n}}$$\n",
    "- **Caso 2** datos normales con varianza desconocida.\n",
    "$$a_n=\\bar{X}- t_{1-\\alpha/2,n-1}\\frac{S}{\\sqrt{n}}\\qquad \\text{y}\\qquad b_n=\\bar{X}+ t_{1-\\alpha/2,n-1}\\frac{S}{\\sqrt{n}}$$\n",
    "- **Caso 3** datos de cualquier distribucion pero tamano de muestra grande.\n",
    "$$a_n=\\bar{X}- z_{1-\\alpha/2}\\frac{S}{\\sqrt{n}}\\qquad \\text{y}\\qquad b_n=\\bar{X}+ z_{1-\\alpha/2}\\frac{S}{\\sqrt{n}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para los casos 1 y 3 ocuparemos la distribución normal y para el caso 2\n",
    "usaremos la distribución t-student"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Usamos la librería **stats** del paquete **scipy**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Puntaje    582.5\n",
       "dtype: float64"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "punt = df_punt[[\"Puntaje\"]] # <== Selección de columna a utilizar\n",
    "punt.mean() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([532.7761846]), array([632.2238154]))"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha=0.2\n",
    "confidence=1 - alpha   # <== error\n",
    "n=len(punt)\n",
    "\n",
    "stats.norm.interval(confidence,punt.mean(),stats.sem(punt))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Lo mismo pero hecho a mano"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Puntaje    532.776185\n",
       "dtype: float64"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Calculamos el límite inferior \n",
    "LI=punt.mean()-stats.norm.ppf(1-alpha/2)*stats.sem(punt)\n",
    "LI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Puntaje    632.223815\n",
       "dtype: float64"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Calculamos el límite superior\n",
    "LS=punt.mean()+stats.norm.ppf(1-alpha/2)*stats.sem(punt)\n",
    "LS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Varianza"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Puntaje    24086.666667\n",
       "dtype: float64"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#La varianza de los puntajes esta dada por\n",
    "punt.var()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El intervalo de confianza para la varianza esta dado por \n",
    "$$a_n=\\frac{(n-1)S^2}{\\chi^2_{1-\\frac{\\alpha}{2},n-1}}\\qquad\\text{y}\\qquad b_n=\\frac{(n-1)S^2}{\\chi^2_{\\frac{\\alpha}{2},n-1}}$$\n",
    "donde \n",
    "- $\\chi^2_{1-\\frac{\\alpha}{2},n-1}$ se calcula usando $\\texttt{stats.chi2.ppf(1-alpha/2,n-1)}$\n",
    "-$\\chi^2_{\\frac{\\alpha}{2},n-1}$ se calcula usando $\\texttt{stats.chi2.ppf(alpha/2,n-1)}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Seteamos la confianza y la significancia\n",
    "alpha=0.1\n",
    "confianza=1-alpha\n",
    "S2=punt.var()\n",
    "n=len(punt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Puntaje    14454.434046\n",
       "dtype: float64"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Calculamos el limite inferior\n",
    "LI=(n-1)*S2/stats.chi2.ppf(1-alpha/2,n-1)\n",
    "LI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Puntaje    49759.370627\n",
       "dtype: float64"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Calculamos el limite superior\n",
    "LS=(n-1)*S2/stats.chi2.ppf(alpha/2,n-1)\n",
    "LS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Visualización** "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "from plotnine import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python3.8/dist-packages/plotnine/stats/stat_bin.py:95: PlotnineWarning: 'stat_bin()' using 'bins = 4'. Pick better value with 'binwidth'.\n"
     ]
    },
    {
     "data": {
      "image/png": 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fujMyMvTSSy+5bS8gIECDBw/Wnj179Oyzz5bY37k+affv31+S9Je//MVt+q5du7RkyRI1a9asxFVKlREQEKA+ffpo69ateuutt9zmzZw5s8T3T/To0UP169fXk08+qR9++KHE9goLCyvURo899pjb7a+++krvvvuuOnfu7Dqlk5SUpKCgoBJjIHbu3Kk333yz3Ps6m8jIyFJrLu55+WVPy6FDh8p9ue2QIUN07NgxPfroo6XO//mpk/I+F0pzxRVXKDExUX//+9914MAB13Sn01niPpbOfGGcdGbM1enTp0vMz87O1qlTp8rcZ0Xs2rXrrGN91q5dq8zMTLdTUsXH895775UYY+V0OjVz5kxJ/3u+4PxFzwkqbNOmTVq8eLGkM92sxd8Q+80336hly5Zavny5oqKiytxGjx49FBUVpc6dO6tx48bKzc3VsmXLlJWVpYcffti1XPv27SVJf/rTn3THHXcoNDRUqampri5xTx06dEjXX3+9+vXrp8zMTM2dO1d5eXmaPXu2W6/H7373O40dO1Y9evRQv379dPToUb344otKSkoqcW5+6tSpWrNmjUaPHq3Vq1erS5cuCg4O1pYtW7R9+/YS36Pyc0OGDNHixYs1Z84c7d+/Xz169HBdSmyM0fPPP+/1XqSpU6dq1apVGjhwoO6//361aNFC69at0z//+U8lJye7nToJDw/XokWL1KdPH7Vs2VJ33XWXWrRooRMnTmj37t1asWKFpk2bVq5vW42Li9PKlSt18OBBde/e3fW9OKGhoa43H+lMeBg+fLief/553XbbberWrZv279+vuXPn6pJLLtH69esrdfzt27fX/PnzNWHCBLVs2VIBAQHq1auXWrRoodTUVE2fPl05OTm6+OKLtXfvXj3//PNKTk4uVwj7/e9/rw8//FCTJ0/Wf/7zH91www2KjY3VgQMH9Nlnn2nPnj2u3p/yPhdKU6tWLT3zzDPq16+f2rZtq/vuu0916tTRihUrSv2CsyuvvFJTp07Vww8/rNTUVA0aNEgNGzbUjz/+qM2bN+sf//iHtm7d6rWvh09LS1P//v3VqVMnde3a1fVt0Zs2bdIrr7yi4OBgt0uXa9WqpRdeeEG9evVS+/btdffdd6tly5bKysrSP//5T33++efq3r27hgwZ4pX6YDEfXimEGqb40s7iP4fDYaKiokyzZs3Mrbfeal555ZUSl5kaU/oluS+++KLp0aOHadCggQkODjbx8fGmc+fO5tVXXy2x/hNPPGGSkpJMYGCg23ZKu4Tz58q6lDg9Pd0MGzbMxMXFmZCQENOuXTu3yx+LFRYWmnHjxpmEhAQTHBxsLr74YrNgwYKz7js7O9uMGzfOpKSkmODgYBMTE2Patm1rnnvuuRL34y8vkc3LyzOTJk1yW/eWW24x69evL1GXfnGJ6Lm2fTbbt283ffr0MVFRUSYyMtJ0797dfP3116ZLly6lXq65bds2M3ToUNOwYUMTFBRk4uLiTJs2bczYsWPdLmU9m+Lt7t271/Tv399ER0eb8PBw061bN/PFF1+UWD4nJ8fcd999Ji4uzoSGhporr7zS/Otf/yrzUuLSlHbZ8JEjR0z//v1NnTp1jMPhcNvevn37zO23327q1atnQkNDzaWXXmrmz59faruf7bFQWFhonnvuOdOuXTsTGRlpQkNDTZMmTUz//v3dHucVeS6czapVq0zbtm1NSEiIiYuLM8OGDTNHjx496+Nk1apVpmfPnqZu3bomKCjIJCQkmGuvvdY89dRTJi8v75z7S0xMNMnJyedc7ujRo2bGjBmmZ8+epkmTJiYsLMyEhISYpKQkM2TIELNx48ZS19u4caMZPHiwufDCC01gYKCJjIw0V155pXn66ac9uuQaNY/DGC+PnAOAs+jatau+++47t292renmzZune+65Rx9//LGuueYaX5cDnBcYcwIAlXDo0CFJZwYZA/AOxpwAgAe+/fZbrVq1Ss8995ySk5PVrFkzX5cEnDfoOQEAD6xatco1mPYf//gHXwwGeBFjTgAAgFXoOQEAAFYhnAAAAKsQTgAAgFUIJwAAwCqEEwAAYBXCCQAAsEqN/RK2/fv3Kz093ddlAACACoiLi1Pjxo3LXKZGhpP9+/erZcuWys3N9XUpAACgAsLDw7Vt27YyA0qNDCfp6enKzc3V4sWL1bJlS1+X41L8E/OBgTXybvVIdna2JCk6OtrHlVQv2tp/0Nb+g7auetu2bdOdd96p9PT08y+cFGvZsqWuuOIKX5fh4o8P7MzMTElSbGysjyupXrS1/6Ct/QdtbQ8GxAIAAKsQTgAAgFUIJwAAwCqEEwAAYBXCCQAAsArhBAAAWIVwAgAArEI4AQAAViGcAAAAq1gTTpYvX67U1FRFREQoMTFRK1as8HVJAADAB6z4jt6PPvpIo0eP1rJly9ShQwelp6crJyfH12UBAAAfsCKcTJw4URMnTlTHjh0lSfXq1VO9evV8XBUAAPAFn5/WKSoq0vr165WZmamUlBQlJCTorrvucv1SIgAA8C8+7zk5cuSITp8+rWXLlumjjz5SZGSkBg8erNGjR2vBggVlrpudne36RUUb+OMvWh47dszXJZTgyMiQIy+vSvfhj2194qcPDAHV8NPqJixMpm7dKt9PefhjW9v4vK4OtHXVK2/Hg89bIDw8XJI0cuRINWzYUJI0fvx49e3b1225pUuXaunSpZLKf3DwP46MDMW0aCGH0+nrUs47MdW4LxMQoKxvv7UmoACoXj4PJzExMWrUqJEcDkeZyw0aNEiDBg2SJG3YsEFt2rRRdHS0YmNjq6PMcvHH1F3MmnbIyZGcTmn4cKl27SrbjfOn8BMQ4PMzo9UmPz9fkhQaGlq1Ozp+XI6XXlKdkBDJgscVz2v/QVtXvehy9rxa0QJ33323Zs+erZ49eyoiIkLTpk1T7969fV0WarLataWqPP1Q3DPjR+HEBAef+U9YmG8LAXDesyKcjBs3Tunp6WrVqpUCAwN18803a8aMGb4uCwAA+IAV4SQwMFDPPPOMnnnmGV+XAgAAfMx/+qQBAECNQDgBAABWIZwAAACrEE4AAIBVCCcAAMAqhBMAAGAVwgkAALAK4QQAAFiFcAIAAKxCOAEAAFYhnAAAAKsQTgAAgFUIJwAAwCqEEwAAYBXCCQAAsArhBAAAWIVwAgAArEI4AQAAViGcAAAAqxBOAACAVQgnAADAKoQTAABgFcIJAACwCuEEAABYhXACAACsQjgBAABWIZwAAACrEE4AAIBVCCcAAMAqhBMAAGAVwgkAALAK4QQAAFiFcAIAAKxCOAEAAFYhnAAAAKsQTgAAgFUIJwAAwCqEEwAAYBXCCQAAsArhBAAAWIVwAgAArEI4AQAAViGcAAAAqxBOAACAVQgnAADAKoQTAABgFcIJAACwCuEEAABYhXACAACsQjgBAABWCfR1AZVRWFiowsJCX5fhYlMt1aX4mK059sJCBUpyOp2S01llu3FW4bZtVXzMVX7sTqcCJBXu3y9Z8LhyPbYDa/TLZYU4s7IkSYU//Vth4eFSXJzX6qku1ryOVaPqfg0v737859kGoGbIy5MkBXbq5ONCzvDHF8nKxgoTEKCigwdrZECBHWr08y4wMFCBFn6asbGmqlJ8rNYc8091BAQESAFVf9YyoBr2YYviY63yYy4qOvPvwIFSQkLV7qscinuK/Kmt8/PzJUmhoaEVX/n4cTleekmBBQU1trfJmtezalDdr+Hl3Y//tACAmiUyUoqO9nUV/zs96EfhxAQHn/lPWJhvC4Hf8p9nGwAAqBEIJwAAwCqEEwAAYBXCCQAAsArhBAAAWIVwAgAArEI4AQAAViGcAAAAqxBOAACAVQgnAADAKoQTAABgFcIJAACwCuEEAABYhXACAACsQjgBAABWIZwAAACrEE4AAIBVCCcAAMAqhBMAAGAVwgkAALAK4QQAAFiFcAIAAKxCOAEAAFYhnAAAAKsQTgAAgFUIJwAAwCqEEwAAYBXCCQAAsArhBAAAWIVwAgAArEI4AQAAViGcAAAAqxBOAACAVQgnAADAKoQTAABgFcIJAACwCuEEAABYhXACAACsQjgBAABWIZwAAACrEE4AAIBVCCcAAMAqhBMAAGAVwgkAALAK4QQAAFiFcAIAAKxCOAEAAFYhnAAAAKtYFU7S09MVFxen9u3b+7oUAADgI1aFkz/+8Y9q1aqVr8sAAAA+ZE04Wbt2rXbu3Km77rrL16UAAAAfsiKcFBQUaOTIkZozZ44cDoevywEAAD4U6OsCJGnatGm6/vrrdemll+rrr78u93rZ2dnKzMz0ai2OjAw58vI8WrewsFCSFBhoxd1aLU5kZ0uSAqKjfVzJGY7DhxUtKT8/XyY4uMr243Q6JUkBAVbk+2qR5+HzoqIcp04pVFL+qVMy1bTPstDWFePIz1eopKysLDkjI71XVDXwx9fwY8eOVev+sn96zzgXn7fArl27tHDhQm3cuLHM5ZYuXaqlS5dKKv/BVZQjI0MxLVrI8dOLEc4txtcFnE1enlS7tq+rAAB4wOfh5JNPPtEPP/yglJQUSWcSe15eni644ALt2LFDtX96gxk0aJAGDRokSdqwYYPatGmj6OhoxcbGeq+YnBzJ6ZSGD/fojc0fP2Hl5+dLkkJDQ31cyU8OHZJee02htWpJYWFVtht/bOtiYVV4v0qSQkIkSaEhIVXahuVFW1dQQYEkKSYmRvLm63M18Meek2JefS8tQ3Q5e9l93gK33XabbrzxRtftV199VX//+9/19ttvKyoqyjdF1a4teXKaorjHxY9exFynTix4E5EkHT/u6woAAJXk83ASFhbmls6jo6MVFBSkCy64wIdVAQAAX7HuI/6wYcO0bt06X5cBAAB8xLpwAgAA/BvhBAAAWIVwAgAArEI4AQAAViGcAAAAqxBOAACAVQgnAADAKoQTAABgFcIJAACwCuEEAABYhXACAACs4pUf/svNzVV+fn6J6dX1E8wAAOD84XE4McZo6tSpev7553X48OFSlykqKvK4MAAA4J88Pq0zY8YMPf300xoxYoSMMRo/frwmTpyolJQUNWnSRC+++KI36wQAAH7C43Ayf/58TZkyRWPGjJEk9e3bV5MmTdKWLVvUsmVL7dq1y2tFAgAA/+FxOPnuu+902WWXqVatWgoKClJWVtaZDQYE6IEHHtDChQu9VCIAAPAnHoeTunXrKicnR5LUuHFjbdiwwTUvPT1dubm5la8OAAD4HY8HxHbs2FFffPGFevbsqcGDB2vy5Mn64YcfFBQUpBdffFHXXXedN+sEAAB+wuNwMnnyZB08eFCSNG7cOGVlZWnp0qXKy8tT9+7d9eyzz3qtSAAA4D88DifNmzdX8+bNJUkhISGaNWuWZs2a5bXCAACAf+IbYgEAgFUq1HPSu3dvPfXUU2rWrJl69+5d5rIOh0P/+Mc/KlUcAADwPxUKJydOnHB96+vx48flcDiqpCgAAOC/KhROVq9e7fr/mjVrvF0LAAAAY04AAIBdPA4n48eP17333lvqvHvvvVcTJ070uCgAAOC/PA4nS5cu1TXXXFPqvE6dOmnp0qUeFwUAAPyXx+Hk0KFDatSoUanzGjZsqO+//97jogAAgP/yOJzEx8crLS2t1HlpaWmKjY31uCgAAOC/PA4nffv21eTJk7V+/Xq36V988YUeeeQR9evXr9LFAQAA/+Px19dPnTpVn376qTp06KCWLVsqISFBhw4d0rZt23TZZZfp0Ucf9WadAADAT3jccxIdHa1169Zp7ty5at26tSSpdevWeuGFF/T5558rOjraa0UCAAD/4XHPiSQFBwfrnnvu0T333OOtegAAgJ+rVDgp9uOPPyo/P7/E9MaNG3tj8wAAwI94HE4yMjI0atQorVixQqdPn3abZ4yRw+Fw/Q4PAABAeXkcTu6++26tXbtWY8eOVatWrRQcHOzNugAAgJ/yOJysXr1azzzzjIYMGeLNegAAgJ/z+GqdmJgYxcXFebMWAAAAz8PJmDFj9Oyzz6qwsNCb9QAAAD/n8Wmdbdu2aevWrUpOTlaXLl0UExPjNt/hcGjWrFmVrQ8AAPgZj8PJypUrFRBwpuPl448/LjGfcAIAADzhcTjZu3evN+sAAACQVIkxJwAAAFWhUuEkPT1df/7zn3XdddcpJSVFW7ZskSTNmjVL69at80qBAADAv3gcTjZs2KBmzZpp2bJlatiwoXbv3q1Tp05Jkg4ePKgZM2Z4rUgAAOA/PA4nDz74oDp06KCdO3dq/vz5Msa45rVr146eEwAA4BGPB8R+8cUXWrFihYKCgkr8hk58fLx+/PHHShcHAAD8j8c9JxERETp+/Hip8/bv36+6det6XBQAAPBfHoeTHj16aOrUqcrIyHBNczgcysvL06xZs9SzZ0+vFAgAAPyLx+HkiSee0PHjx9WsWTMNHDhQDodDDz/8sFq1aqWMjAxNnTrVm3UCAAA/4fGYkwsvvFAbN27UjBkz9P777ys5OVkZGRm644479Ic//EGxsbHerLNUhYWF3v1tn8JCBUpyOp2S01nh1Z0erFPTFR+zNcdujAIkOY3xqA3Ly5rjrUbV1tbV1IblRVtXeGUF6Mzrs2rYb6/542/FFR9zdR17effjcTiRzvwy8ZQpUzRlypTKbAYAAMClUuHE1wIDAxUY6MVD+GlbAQEBUoDn308XUIl1a5riY7XmmB0OSVKAw1GpNiwva467GlRbW1dzG5YXbV3ulSWdeX2WN1+fq5FX31csV3ys1XXM5d2Px9UkJSXJ8dOLyNns2bPH080DAAA/5XE46dOnT4lwcuzYMa1du1bGGPXv37/SxQEAAP/jcTiZOXNmqdMLCgrUt29fJSUlebppAADgx7x+EjU4OFgjR47Uk08+6e1NAwAAP1AlI7zS09N14sSJqtg0AAA4z3l8WmfFihUlphUUFGjbtm2aPXu2unXrVqnCAACAf/I4nAwYMKDU6UFBQerfv7+effZZj4sCAAD+y+Nwsnfv3hLTQkNDVa9evXNeYgwAAHA2HoeTxMREb9YBAAAgqRIDYl999dWzXpHz17/+Va+//rrHRQEAAP/lcTh5/PHHFRISUuq8sLAwTZs2zeOiAACA//I4nOzcuVOpqamlzmvVqpV27NjhcVEAAMB/eRxOQkNDdeTIkVLnHT582K9+OAkAAHiPx+GkS5cumjZtmk6ePOk2/eTJk5o+fbq6du1a2doAAIAf8rh747HHHlOHDh2UnJysAQMGKCEhQYcOHdIbb7yhgoICLVu2zJt1AgAAP+FxOGnRooW++OILTZo0ScuXL1dGRobq1q2r7t27a9KkSbrooou8WScAAPATlRoYctFFF+mVV17xVi0AAADe+eG/AwcO6LPPPisx/gQAAKCiKhVOXnjhBV144YVKTExUp06dtH37dklSv379NGvWLK8UCAAA/IvH4WTmzJkaNWqUhgwZovfee0/GGNe8rl278g2xAADAIx6POXn22Wc1YcIEPfzwwyoqKnKb17x5c1cvCgAAQEV43HNy8OBBXX311aXOCwoKUk5OjsdFAQAA/+VxOElMTNT69etLnfff//5XKSkpHhcFAAD8l8fh5J577tHUqVM1f/58HT9+XJJ0+vRpvf3223ryySd17733eq1IAADgPzwec/LQQw9p//79+u1vf+sKIh07dpQkPfDAA3rggQe8UyEAAPArlfoStmeeeUajR4/WBx98oPT0dMXGxuq6665Ts2bNvFUfAADwMx6Fk/z8fNWvX1+LFy9Wr1699Nvf/tbbdQEAAD/l0ZiT0NBQhYeHKzCwUh0vAAAAJXg8IHbo0KGaN2+eN2sBAADwfMxJnTp1tG7dOrVu3Vo33XST6tevL4fD4ZrvcDj04IMPeqVIAADgPzwOJ2PHjpUkHT58WFu2bCkxn3ACAAA8UaHTOpdcconS0tIkSU6nU06nU4sWLVJGRobrdvHfL7/SHgAAoDwqFE7S0tKUm5vrul1UVKQhQ4Zo7969Xi8MAAD4J48HxBb7+a8RAwAAVFalwwkAAIA3VTic/PyKnLKmAQAAeKLCV+tce+21CghwzzSdOnUqMc3hcCg7O7ty1QEAAL9ToXAyadKkqqoDAABAEuEEAABYhgGxAADAKoQTAABgFcIJAACwCuEEAABYhXACAACsQjgBAABWIZwAAACrEE4AAIBVCCcAAMAqhBMAAGAVn4eTU6dO6e6771ZSUpKioqJ08cUXa8mSJb4uCwAA+EiFf5XY2woLC5WQkKAPP/xQSUlJ+vTTT3XzzTcrKSlJHTp08HV5AACgmvm85yQiIkKPPPKImjZtKofDoWuuuUYdO3bUZ5995uvSAACAD/g8nPzSyZMn9eWXXyo1NdXXpQAAAB/w+Wmdn3M6nRo2bJiuuuoq3XDDDedcPjs7W5mZmV7bf0BWlmIk5efnywQHV3h9p9N5ZjsB1mW+KpOXl+frEtw4Tp1SqKT8U6dkqrA22rrqVFcblhdtXTGO/HyFSsrKypIzMtJ7RVWDwsJCSVJgoFVvjVXq2LFj1bq/7Ozsci1nTQsYY3Tffffp0KFDevfdd+VwONzmL126VEuXLpVU/oMDAAA1jxXhxBijESNGaOPGjfrggw8UWUraHjRokAYNGiRJ2rBhg9q0aaPo6GjFxsZ6r5CcHElSaGioFBZW4dX98RNWsTAP7q8qERIiSQoNCfGoDcuLtq5C1dSG5UVbV1BBgSQpJiZG8ubrczXwx56TYl59Ly1DdHR0uZazogVGjhypdevW6cMPP1Tt2rV9XQ4AAPAhn38U2Ldvn5577jlt3bpVjRo1UmRkpCIjI/XYY4/5ujQAAOADPu85SUxMlDHG12UAAABL+LznBAAA4OcIJwAAwCqEEwAAYBXCCQAAsArhBAAAWIVwAgAArEI4AQAAViGcAAAAqxBOAACAVQgnAADAKoQTAABgFcIJAACwCuEEAABYhXACAACsQjgBAABWIZwAAACrEE4AAIBVCCcAAMAqhBMAAGAVwgkAALAK4QQAAFiFcAIAAKxCOAEAAFYhnAAAAKsQTgAAgFUIJwAAwCqEEwAAYBXCCQAAsArhBAAAWIVwAgAArEI4AQAAViGcAAAAqxBOAACAVQgnAADAKoQTAABgFcIJAACwCuEEAABYhXACAACsQjgBAABWIZwAAACrEE4AAIBVCCcAAMAqhBMAAGAVwgkAALAK4QQAAFiFcAIAAKxCOAEAAFYhnAAAAKsQTgAAgFUIJwAAwCqBvi6gMgoLC1VYWOjNDSpQktPplJzOCq/u9GCdmq74mK05dmMUIMlpjEdtWF7WHG81qra2rqY2LC/ausIrK0BS4f79kjdfn6uB6/0ksEa/NVaIMytLklT4078u4eFSXJzX91fe92z/aQEAQNXLy5MkBXbq5ONCKs4f3xDPFj9MQICKDh6skoBSHjW6LQIDAxXozYT707YCAgKkAM/PeAVUYt2apvhYrTlmh0OSFOBwVKoNy8ua464G1dbW1dyG5UVbl1NR0Zl/Bw6UEhK8WFXVK+4p8qe2zs/PlySFhob+b+Lx43K89JICCwq83otU3vfsGh1OAACWioyUoqN9XUXFFJ/G8qNwYoKDz/wnLMy3hfyC/7QAAACoEQgnAADAKoQTAABgFcIJAACwCuEEAABYhXACAACsQjgBAABWIZwAAACrEE4AAIBVCCcAAMAqhBMAAGAVwgkAALAK4QQAAFiFcAIAAKxCOAEAAFYhnAAAAKsQTgAAgFUIJwAAwCqEEwAAYBXCCQAAsArhBAAAWIVwAgAArEI4AQAAViGcAAAAqxBOAACAVQgnAADAKoQTAABgFcIJAACwCuEEAABYhXACAACsQjgBAABWIZwAAACrEE4AAIBVCCcAAMAqhBMAAGAVwgkAALAK4QQAAFiFcAIAAKxCOAEAAFYhnAAAAKsQTgAAgFUIJwAAwCqEEwAAYBXCCQAAsArhBAAAWIVwAgAArEI4AQAAViGcAAAAq1gRTrKysjRw4EBFRUUpISFBM2fO9HVJAADARwJ9XYAkjRw5UqdOndLBgwe1b98+XXfddWrevLluuukmX5cGAACqmc97Tk6ePKnXX39djz76qGrXrq3WrVvrnnvu0UsvveTr0gAAgA/4PJzs2LFDTqdTqamprmmXXXaZ0tLSfFgVAADwFZ+f1snJyVF0dLTbtJiYGJ04ceKc62ZnZyszM9NrtQRkZSlG0qmjR2Xy8yu8vtPpPLOdAJ9nvmpz6qf7yREa6uNKznBkZipE0qnMTJng4CrbD21ddaqrDcuLtq4Y29qvImjrMxwnTihEZ8aDOiMjvbq/7Ozsci3n83ASGRmp48ePu03Lzs5WVFSU27SlS5dq6dKlrvlVwYSFyQQEKOSVV6pk++cjOyKJOyMp5K23fF3Geac625o29K3KtjXtV3Ocra1NQIBMWFi11vJzPg8nKSkpcjgc2rJliy6++GJJ0saNG91O80jSoEGDNGjQIEnShg0b1KZNG0VHRys2NtZ7xcTGSkeOSLm5Hq1eWFgoSQoM9PndWm2ysrIknentsoUjL0+q4icVbV21qqMNy4u2rjib2q8iaOv/cYSHq05cnNf398szJWfj8xaIiIjQgAEDNH78eC1atEj79u3TvHnztGDBAt8UVJnG+OmBLT96YLu6/LwZEmsC2tp/0Nb+g7a2hhUn1ubMmaOgoCA1aNBA3bt315///GcuIwYAwE9ZEQ9jYmL0+uuv+7oMAABgASt6TgAAAIoRTgAAgFUIJwAAwCqEEwAAYBXCCQAAsArhBAAAWIVwAgAArEI4AQAAViGcAAAAqxBOAACAVQgnAADAKlb8to6ntm3b5usS3Pjjz21nZ2dLKv/PYJ8vaGv/QVv7D9q66pX3fbtGtkBcXJzCw8N15513+roUAABQAeHh4YqLiytzGYcxxlRTPV61f/9+paen+7oMSHrwwQc1Y8YMX5eBakBb+w/a2n9Ud1vHxcWpcePGZS5TI3tOJKlx48bnPDhUj+joaF1xxRW+LgPVgLb2H7S1/7CxrRkQCwAArEI4QaUNGjTI1yWgmtDW/oO29h82tnWNHXMCAADOT/ScAAAAqxBOAACAVQgnOKf09HTFxcWpffv2rmlpaWlq3769wsPD1apVK3300Udu67zxxhtKTk5WeHi4rrvuOu3bt6+6y0YFLV++XKmpqYqIiFBiYqJWrFghibY+33z33Xe6+eabFRsbq3r16mnIkCE6ceKEJOnAgQO64YYbFBERoaSkJC1btsxt3bVr1yo1NVXh4eG66qqrtGnTJl8cAs5i9uzZuvLKKxUSEqLbb7/dbV5ln8cTJkxQfHy8oqOjdffdd+vUqVNVezAGOIdhw4aZTp06mXbt2hljjCkoKDBNmjQxjz76qMnPzzfLli0ztWvXNkeOHDHGGLN161YTERFh3n33XZObm2tGjx5t2rZt68tDwDl8+OGHpmHDhuaTTz4xRUVF5siRI2b37t209Xmoe/fu5s477zS5ubkmMzPTXHvtteb3v/+9McaYjh07mvvvv9/k5uaa1atXm8jISLN582ZjjDHp6ekmOjravPzyyyY/P9889dRTpnHjxiY/P9+HR4OfW758uXnzzTfNiBEjzG233eaaXtnn8YsvvmiSkpLM7t27TXp6uunYsaMZM2ZMlR4L4QRlWrNmjenYsaN56aWXXOHkvffeM/Xq1TNFRUWu5a6++moze/ZsY4wx48aNM7/61a9c844fP25CQkJMWlpa9RaPcuvYsaN54YUXSkynrc8/F110kXn77bddt2fPnm26detmduzYYQIDA01mZqZr3uDBg81DDz1kjDHmhRdeMG3atHHNczqdpmHDhmblypXVVzzKZdKkSW7hpLLP46uvvto8++yzbtuLj4+v0mPgtA7OqqCgQCNHjtScOXPkcDhc09PS0tS6dWsFBPzv4XPZZZcpLS3NNf+yyy5zzYuKilJycrJrPuxSVFSk9evXKzMzUykpKUpISNBdd92l7Oxs2vo8NHr0aC1ZskQ5OTlKT0/X66+/rptuuklpaWlKTExUnTp1XMuW1dYOh0OXXHIJbV0DVPZ5/Mv5l112mY4ePaojR45UWc2EE5zVtGnTdP311+vSSy91m56Tk6OYmBi3aTExMa7z1ueaD7scOXJEp0+f1rJly/TRRx9p69atOnLkiEaPHk1bn4e6du2qb7/9VtHR0YqPj1dISIh+97vf0dbnscq27S/nF/+/KtuecIJS7dq1SwsXLtSUKVNKzIuMjHT9kmWx7OxsRUVFlWs+7BIeHi5JGjlypBo2bKiYmBiNHz9eK1eupK3PM0VFRbrxxht188036+TJk8rOztaFF16oO++8k7Y+j1W2bX85v/j/Vdn2hBOU6pNPPtEPP/yglJQUXXDBBfr973+vDRs26IILLlDTpk21efNmOZ1O1/IbN25UamqqJCk1NVUbN250zcvJydHu3btd82GXmJgYNWrUyO3UXbHU1FTa+jxy7Ngxff/99xo1apRCQ0NVu3Zt3X///XrnnXeUmpqqffv2KSsry7V8WW1tjNE333xDW9cAlX0e/3L+xo0bFR8fr/r161dd0VU6ogU1Vm5urjl8+LDrb+bMmeaKK64whw8fdo38fvzxx01+fr557bXXSh35/f7775u8vDzzhz/8gSs4LDdlyhRz+eWXm8OHD5vjx4+bW265xQwfPpy2Pg81bdrUTJkyxZw6dcrk5OSYu+++21x99dXGmDMDH0eMGGFyc3PN2rVrTVRUVImrdRYtWmROnTplZsyYYRo1asTVOhY5ffq0ycvLM+PHjze33nqrycvLMwUFBZV+Hr/wwgsmOTnZ7Nmzx2RkZJhOnTpxtQ7ssGDBAtfVOsYY880335i2bdua0NBQ06JFC/PBBx+4Lf/aa6+ZpKQkExYWZq699lrz3XffVXfJqIDTp0+bUaNGmTp16pj4+HgzbNgwk52dbYyhrc83mzZtMt26dTN16tQxsbGxpmfPnmbXrl3GGGP2799vrr/+ehMWFmYSExPNkiVL3NZdvXq1ufjii01oaKi58sorzddff+2DI8DZTJo0yUhy+xs6dKgxpnLPY6fTacaPH2/q1q1rateubYYPH17loZTf1gEAAFZhzAkAALAK4QQAAFiFcAIAAKxCOAEAAFYhnAAAAKsQTgAAgFUIJwAAwCqEEwAAYBXCCQBJ0uTJk+VwOFx/8fHx6tatmz7++GOv72vjxo2aPHmycnNzPVrf4XDor3/9q5erAmALwgkAl7CwMH3++ef6/PPP9be//U0ZGRm67rrrlJaW5tX9bNy4UVOmTPE4nHz++ee64447vFoTAHsE+roAAPYICAhQ+/btXbfbtm2rJk2aaO7cuZo9e7YPK3P38xoBnH/oOQFwVo0bN1Z8fLz27t2rrl276pZbbnGbv3HjRjkcDq1Zs8Y1zeFwaPr06Zo8ebLq16+vuLg43XXXXTp58qQkaeHChbrrrrskSfHx8XI4HGrSpIkk6fDhwxo+fLiaNm2qsLAwNWvWTOPGjdOpU6fc9lvaaZ23335b7dq1U1hYmOLj43X//fe79gmgZiGcADir48ePKyMjQwkJCRVab/bs2dq5c6defvllTZw4UUuWLNFf/vIXSdLNN9+shx9+WJK0atUqff7553rzzTclSenp6YqNjdXTTz+tVatWacyYMXr55Zd13333lbm/N954Q71791br1q315ptvavr06VqxYoV+85vfeHDUAHyN0zoA3BQWFkqSvv/+e/3f//2fioqKNGDAAD3++OPl3kaDBg30yiuvSJJuvPFGbdiwQW+88YamTZum+Ph4JScnS5LatGmjuLg413qtW7d26xHp2LGjIiIiNHToUM2ZM0fh4eEl9mWM0UMPPaTbbrtN8+bNc6uhZ8+emjBhgi6++OKK3QkAfIqeEwAuJ0+eVFBQkIKCgpSUlKTVq1dr9uzZ6tGjR4W20717d7fbrVq10vfff3/O9Ywxmjlzplq1aqWwsDAFBQXpjjvuUGFhofbs2VPqOjt27NC+ffs0cOBAFRYWuv66dOmigIAAffnllxWqHYDv0XMCwCUsLEz/+c9/5HA4FBcXp0aNGikgoOKfYWJiYtxuBwcHlxg3UpqZM2fqoYce0pgxY3TttdeqTp06+uKLLzRixAjl5+eXuk56erokqV+/fqXOP3DgQMWKB+BzhBMALgEBAbryyitLnRcaGqqCggK3aceOHfPq/l9//XX17t3b7RTS1q1by1wnNjZW0plxLu3atSsxv6LjZQD4HuEEQLk0bNhQ77//vowxcjgckqT33nvPo20FBwdLUonekLy8PNe8YsVjV86mRYsWatiwofbs2aMRI0Z4VA8AuxBOAJTLgAEDNH/+fI0aNUp9+/bVZ599pjfeeMOjbbVs2VKSNGfOHPXt21fh4eFq3bq1unfvrlmzZmn27NlKSUnR4sWLtWvXrjK35XA49PTTT2vw4ME6efKkbr75ZkVERGjfvn16++239dhjjyklJcWjOgH4BgNiAZTLjTfeqOnTp+uf//yn+vbtq7S0NM2dO9ejbV1++eWaPHmyFi9erKuvvlq9evWSJE2cOFGDBw/WxIkTdfvttys0NFTPPPNMqdso7r2RpFtvvVXvvPOOvv32Ww0aNEi9e/fWU089pSZNmqh+/foe1QjAdxzGGOPrIgCgvE6cOKHatWvrpZdecn2ZG4DzC6d1ANQY3377rf7+979L0lkH7gKo+QgnAGqMsWPH6r///a+eeOIJtW7d2tflAKginNYBAABWYUAsAACwCuEEAABYhXACAACsQjgBAABWIZwAAACrEE4AAIBVCCcAAMAqhBMAAGAVwgkAALDK/wOs7hA63Wuz6QAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<ggplot: (8791333533937)>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(ggplot(punt)+\n",
    " aes(x=\"Puntaje\")+\n",
    " geom_histogram(color=\"red\",fill=\"red\",alpha=0.5)+ \n",
    " theme_linedraw()+\n",
    " ggtitle(\"Distribución de puntajes de PSU\")+\n",
    " xlab(\"Puntaje\")+\n",
    " ylab(\"Frecuencia\")\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Test de hipótesis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "• Una **hipótesis** estadística es una afirmación sobre los parámetros de una o más \n",
    "poblaciones. En vez de hacer una estimación de un parámetro se hace una afirmación."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "• Utilizaremos la información proporcionada de una muestra aleatoria para decidir si \n",
    "esta afirmación es verdadera o falsa."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "• En sí, nos es utíl para ver la **veracidad** de una idea (afirmación) que querramos comprobar y para ver \"resultado final\", aplicamos técnicas estadísticas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "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>Nombre</th>\n",
       "      <th>Sexo</th>\n",
       "      <th>Edad</th>\n",
       "      <th>Puntaje</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Anita</td>\n",
       "      <td>M</td>\n",
       "      <td>17</td>\n",
       "      <td>630</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Bastian</td>\n",
       "      <td>H</td>\n",
       "      <td>19</td>\n",
       "      <td>450</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Carlos</td>\n",
       "      <td>H</td>\n",
       "      <td>17</td>\n",
       "      <td>550</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Andrea</td>\n",
       "      <td>M</td>\n",
       "      <td>17</td>\n",
       "      <td>680</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Ignacia</td>\n",
       "      <td>M</td>\n",
       "      <td>20</td>\n",
       "      <td>710</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>Antonio</td>\n",
       "      <td>H</td>\n",
       "      <td>17</td>\n",
       "      <td>450</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>Nicholas</td>\n",
       "      <td>H</td>\n",
       "      <td>21</td>\n",
       "      <td>600</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>Vicente</td>\n",
       "      <td>H</td>\n",
       "      <td>19</td>\n",
       "      <td>630</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>Diego</td>\n",
       "      <td>H</td>\n",
       "      <td>20</td>\n",
       "      <td>700</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>Francisca</td>\n",
       "      <td>M</td>\n",
       "      <td>20</td>\n",
       "      <td>550</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>Antonia</td>\n",
       "      <td>M</td>\n",
       "      <td>20</td>\n",
       "      <td>650</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>Catalina</td>\n",
       "      <td>M</td>\n",
       "      <td>21</td>\n",
       "      <td>310</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>Maria</td>\n",
       "      <td>M</td>\n",
       "      <td>18</td>\n",
       "      <td>300</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>Carolina</td>\n",
       "      <td>M</td>\n",
       "      <td>20</td>\n",
       "      <td>480</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>Matias</td>\n",
       "      <td>H</td>\n",
       "      <td>18</td>\n",
       "      <td>780</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>Joaquin</td>\n",
       "      <td>H</td>\n",
       "      <td>21</td>\n",
       "      <td>850</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       Nombre Sexo  Edad  Puntaje\n",
       "0       Anita    M    17      630\n",
       "1     Bastian    H    19      450\n",
       "2      Carlos    H    17      550\n",
       "3      Andrea    M    17      680\n",
       "4     Ignacia    M    20      710\n",
       "5     Antonio    H    17      450\n",
       "6    Nicholas    H    21      600\n",
       "7     Vicente    H    19      630\n",
       "8       Diego    H    20      700\n",
       "9   Francisca    M    20      550\n",
       "10    Antonia    M    20      650\n",
       "11   Catalina    M    21      310\n",
       "12      Maria    M    18      300\n",
       "13   Carolina    M    20      480\n",
       "14     Matias    H    18      780\n",
       "15    Joaquin    H    21      850"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_punt # <== data a utilizar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "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>Nombre</th>\n",
       "      <th>Sexo</th>\n",
       "      <th>Edad</th>\n",
       "      <th>Puntaje</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Anita</td>\n",
       "      <td>M</td>\n",
       "      <td>17</td>\n",
       "      <td>630</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Bastian</td>\n",
       "      <td>H</td>\n",
       "      <td>19</td>\n",
       "      <td>450</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Carlos</td>\n",
       "      <td>H</td>\n",
       "      <td>17</td>\n",
       "      <td>550</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Andrea</td>\n",
       "      <td>M</td>\n",
       "      <td>17</td>\n",
       "      <td>680</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Ignacia</td>\n",
       "      <td>M</td>\n",
       "      <td>20</td>\n",
       "      <td>710</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>Antonio</td>\n",
       "      <td>H</td>\n",
       "      <td>17</td>\n",
       "      <td>450</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>Nicholas</td>\n",
       "      <td>H</td>\n",
       "      <td>21</td>\n",
       "      <td>600</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>Vicente</td>\n",
       "      <td>H</td>\n",
       "      <td>19</td>\n",
       "      <td>630</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     Nombre Sexo  Edad  Puntaje\n",
       "0     Anita    M    17      630\n",
       "1   Bastian    H    19      450\n",
       "2    Carlos    H    17      550\n",
       "3    Andrea    M    17      680\n",
       "4   Ignacia    M    20      710\n",
       "5   Antonio    H    17      450\n",
       "6  Nicholas    H    21      600\n",
       "7   Vicente    H    19      630"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "muestra = df_punt.iloc[0:8]\n",
    "muestra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Test para la media**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "puntajes = df_punt[[\"Puntaje\"]]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "media=puntajes.mean()\n",
    "ds=puntajes.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ttest_1sampResult(statistic=array([2.12630514]), pvalue=array([0.0504884]))"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Podemos concluir que el parametro de puntajes es igual a 500 puntos?\n",
    "#h0=500\n",
    "#h1!=500\n",
    "stats.ttest_1samp(puntajes,500,alternative='two-sided')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[8.72780496e-05]\n",
      "[0.0067714]\n",
      "[0.11295651]\n",
      "[0.42507385]\n",
      "[0.37456796]\n",
      "[0.080543]\n"
     ]
    }
   ],
   "source": [
    "#Definimos 6 regiones: (-Inf,0),(0,200),(200,400),(400,600),(600,800),(800,Inf)\n",
    "p1=stats.norm.cdf(0,loc=media,scale=ds)\n",
    "print(p1)\n",
    "p2=stats.norm.cdf(200,loc=media,scale=ds)-stats.norm.cdf(0,loc=media,scale=ds)\n",
    "print(p2)\n",
    "p3=stats.norm.cdf(400,loc=media,scale=ds)-stats.norm.cdf(200,loc=media,scale=ds)\n",
    "print(p3)\n",
    "p4=stats.norm.cdf(600,loc=media,scale=ds)-stats.norm.cdf(400,loc=media,scale=ds)\n",
    "print(p4)\n",
    "p5=stats.norm.cdf(800,loc=media,scale=ds)-stats.norm.cdf(600,loc=media,scale=ds)\n",
    "print(p5)\n",
    "p6=1-stats.norm.cdf(800,loc=media,scale=ds)\n",
    "print(p6)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Calculamos el tamano de muestra\n",
    "n=len(muestra)\n",
    "n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[array([0.00069822]),\n",
       " array([0.0541712]),\n",
       " array([0.90365206]),\n",
       " array([3.40059076]),\n",
       " array([2.99654371]),\n",
       " array([0.64434403])]"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Calculamos las frecuencias esperadas\n",
    "E1=n*p1\n",
    "E2=n*p2\n",
    "E3=n*p3\n",
    "E4=n*p4\n",
    "E5=n*p5\n",
    "E6=n*p6\n",
    "\n",
    "FE=[E1,E2,E3,E4,E5,E6]\n",
    "FE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0, 16,  0,  0,  0,  0])"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FO=pd.cut(pd.array(Edad), bins=[-np.inf,0,200,400,600,800,np.inf]).value_counts()\n",
    "FO=np.array(FO)\n",
    "FO"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "For each axis slice, the sum of the observed frequencies must agree with the sum of the expected frequencies to a relative tolerance of 1e-08, but the percent differences are:\n[inf 11. inf inf inf inf]",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "Cell \u001b[0;32mIn [70], line 2\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[39m#Aplicamos el test\u001b[39;00m\n\u001b[0;32m----> 2\u001b[0m stats\u001b[39m.\u001b[39;49mchisquare(FO,FE)\n",
      "File \u001b[0;32m/usr/local/lib/python3.8/dist-packages/scipy/stats/_stats_py.py:7305\u001b[0m, in \u001b[0;36mchisquare\u001b[0;34m(f_obs, f_exp, ddof, axis)\u001b[0m\n\u001b[1;32m   7181\u001b[0m \u001b[39mdef\u001b[39;00m \u001b[39mchisquare\u001b[39m(f_obs, f_exp\u001b[39m=\u001b[39m\u001b[39mNone\u001b[39;00m, ddof\u001b[39m=\u001b[39m\u001b[39m0\u001b[39m, axis\u001b[39m=\u001b[39m\u001b[39m0\u001b[39m):\n\u001b[1;32m   7182\u001b[0m     \u001b[39m\"\"\"Calculate a one-way chi-square test.\u001b[39;00m\n\u001b[1;32m   7183\u001b[0m \n\u001b[1;32m   7184\u001b[0m \u001b[39m    The chi-square test tests the null hypothesis that the categorical data\u001b[39;00m\n\u001b[0;32m   (...)\u001b[0m\n\u001b[1;32m   7303\u001b[0m \n\u001b[1;32m   7304\u001b[0m \u001b[39m    \"\"\"\u001b[39;00m\n\u001b[0;32m-> 7305\u001b[0m     \u001b[39mreturn\u001b[39;00m power_divergence(f_obs, f_exp\u001b[39m=\u001b[39;49mf_exp, ddof\u001b[39m=\u001b[39;49mddof, axis\u001b[39m=\u001b[39;49maxis,\n\u001b[1;32m   7306\u001b[0m                             lambda_\u001b[39m=\u001b[39;49m\u001b[39m\"\u001b[39;49m\u001b[39mpearson\u001b[39;49m\u001b[39m\"\u001b[39;49m)\n",
      "File \u001b[0;32m/usr/local/lib/python3.8/dist-packages/scipy/stats/_stats_py.py:7147\u001b[0m, in \u001b[0;36mpower_divergence\u001b[0;34m(f_obs, f_exp, ddof, axis, lambda_)\u001b[0m\n\u001b[1;32m   7141\u001b[0m     \u001b[39mif\u001b[39;00m diff_gt_tol:\n\u001b[1;32m   7142\u001b[0m         msg \u001b[39m=\u001b[39m (\u001b[39mf\u001b[39m\u001b[39m\"\u001b[39m\u001b[39mFor each axis slice, the sum of the observed \u001b[39m\u001b[39m\"\u001b[39m\n\u001b[1;32m   7143\u001b[0m                \u001b[39mf\u001b[39m\u001b[39m\"\u001b[39m\u001b[39mfrequencies must agree with the sum of the \u001b[39m\u001b[39m\"\u001b[39m\n\u001b[1;32m   7144\u001b[0m                \u001b[39mf\u001b[39m\u001b[39m\"\u001b[39m\u001b[39mexpected frequencies to a relative tolerance \u001b[39m\u001b[39m\"\u001b[39m\n\u001b[1;32m   7145\u001b[0m                \u001b[39mf\u001b[39m\u001b[39m\"\u001b[39m\u001b[39mof \u001b[39m\u001b[39m{\u001b[39;00mrtol\u001b[39m}\u001b[39;00m\u001b[39m, but the percent differences are:\u001b[39m\u001b[39m\\n\u001b[39;00m\u001b[39m\"\u001b[39m\n\u001b[1;32m   7146\u001b[0m                \u001b[39mf\u001b[39m\u001b[39m\"\u001b[39m\u001b[39m{\u001b[39;00mrelative_diff\u001b[39m}\u001b[39;00m\u001b[39m\"\u001b[39m)\n\u001b[0;32m-> 7147\u001b[0m         \u001b[39mraise\u001b[39;00m \u001b[39mValueError\u001b[39;00m(msg)\n\u001b[1;32m   7149\u001b[0m \u001b[39melse\u001b[39;00m:\n\u001b[1;32m   7150\u001b[0m     \u001b[39m# Ignore 'invalid' errors so the edge case of a data set with length 0\u001b[39;00m\n\u001b[1;32m   7151\u001b[0m     \u001b[39m# is handled without spurious warnings.\u001b[39;00m\n\u001b[1;32m   7152\u001b[0m     \u001b[39mwith\u001b[39;00m np\u001b[39m.\u001b[39merrstate(invalid\u001b[39m=\u001b[39m\u001b[39m'\u001b[39m\u001b[39mignore\u001b[39m\u001b[39m'\u001b[39m):\n",
      "\u001b[0;31mValueError\u001b[0m: For each axis slice, the sum of the observed frequencies must agree with the sum of the expected frequencies to a relative tolerance of 1e-08, but the percent differences are:\n[inf 11. inf inf inf inf]"
     ]
    }
   ],
   "source": [
    "#Aplicamos el test\n",
    "stats.chisquare(FO,FE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Links con más info"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://relopezbriega.github.io/blog/2015/06/27/probabilidad-y-estadistica-con-python/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ejercicios"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1.- Cree una base de datos (dataset) a mano y calcule el intervalo de confianza de alguna variable.\n",
    "* Hint: puedes aplicar o transformar un diccionario hecho a mano y cambiarlo de formato a DataFrame (como los datos PSU del principio)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Escriba su solución acá"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.- Sobre el anterior, cree una hipotesis a refutar y comente al respecto.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Escriba su solución acá"
   ]
  }
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