Control4.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "set.seed(0)\n",
    "X1 = rnorm(100,5,2)\n",
    "X2 = rnorm(100,3,2)\n",
    "X3 = rexp(100,10)\n",
    "Y  = 2*rep(1,100) + 4*X1 + 3*X2 + rnorm(100,0,2)\n",
    "tiempo = seq(1,100)\n",
    "T = 5*exp(tiempo*(-0.1)+ rnorm(100,0,0.3)) \n",
    "# T = Aexp(Bt)\n",
    "# Log(T) = Log(A) + Bt\n",
    "data = data.frame(X1,X2,X3,Y,tiempo,T)\n",
    "write.csv(data,'base_control_4.csv',row.names=FALSE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "The following objects are masked _by_ .GlobalEnv:\n",
      "\n",
      "    T, tiempo, X1, X2, X3, Y\n",
      "\n",
      "\n",
      "The following objects are masked from data (pos = 3):\n",
      "\n",
      "    T, tiempo, X1, X2, X3, Y\n",
      "\n",
      "\n",
      "The following object is masked from package:base:\n",
      "\n",
      "    T\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "data=read.csv('base_control_4.csv')\n",
    "attach(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Realice una regresión lineal simple con la estructura $$Y_i = \\beta_0 + \\beta_1 X1_i + \\epsilon_i$$ Obtenga el limite inferior de un intervalo de confianza para $$\\hat{\\beta_0}$$ con una confianza de 0.95."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A matrix: 2 × 2 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>2.5 %</th><th scope=col>97.5 %</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>(Intercept)</th><td>4.944154</td><td>12.532447</td></tr>\n",
       "\t<tr><th scope=row>X1</th><td>3.713205</td><td> 5.133613</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 2 × 2 of type dbl\n",
       "\\begin{tabular}{r|ll}\n",
       "  & 2.5 \\% & 97.5 \\%\\\\\n",
       "\\hline\n",
       "\t(Intercept) & 4.944154 & 12.532447\\\\\n",
       "\tX1 & 3.713205 &  5.133613\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 2 × 2 of type dbl\n",
       "\n",
       "| <!--/--> | 2.5 % | 97.5 % |\n",
       "|---|---|---|\n",
       "| (Intercept) | 4.944154 | 12.532447 |\n",
       "| X1 | 3.713205 |  5.133613 |\n",
       "\n"
      ],
      "text/plain": [
       "            2.5 %    97.5 %   \n",
       "(Intercept) 4.944154 12.532447\n",
       "X1          3.713205  5.133613"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "reg1=lm(Y~X1)\n",
    "confint.lm(reg1)\n",
    "# 4.944154"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Considere el modelo de regresión multiple $$Y_i = \\beta_0 + \\beta_1 X1_i + \\beta_2 X2_i + \\epsilon_i$$ Suponga que se desea estimar cuanto vale Y si X1 = 2 y X2 = 3. Obtenga la estimación de Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(Intercept) \n",
      "   19.18352 \n"
     ]
    }
   ],
   "source": [
    "reg2 = lm(Y~X1+X2)\n",
    "estimacion_Y = reg2$coef[1] + reg2$coef[2]*2 + reg2$coef[3]*3\n",
    "print(estimacion_Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Considere el siguiente modelo de regresión multiple\n",
    "$$\n",
    "Y_i = \\beta_0 + \\beta_1 X1_i + \\beta_2 X2_i + \\beta_3 X3_i\n",
    "$$\n",
    "Realice el test de hipotesis $$H_0: \\beta_3 = 0$$ v/s $$H_1:\\beta_3 \\neq 0$$ y entregue como resultado el valor p (p-valor) asocioado al test."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Y ~ X1 + X2 + X3)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-4.9371 -1.1715 -0.1637  1.3314  3.6559 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)   1.7601     0.6544   2.690  0.00843 ** \n",
       "X1            3.9916     0.1105  36.119  < 2e-16 ***\n",
       "X2            3.1115     0.1010  30.796  < 2e-16 ***\n",
       "X3            0.9946     1.7512   0.568  0.57140    \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 1.925 on 96 degrees of freedom\n",
       "Multiple R-squared:  0.9641,\tAdjusted R-squared:  0.963 \n",
       "F-statistic: 858.9 on 3 and 96 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "reg3 = lm(Y~X1+X2+X3)\n",
    "summary(reg3)\n",
    "# 0.57140"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En la base de datos, el par (tiempo,T), corresponde a la temperatura de un cuerpo en el tiempo indicado. Un experto dice que esto se modela mediante la ley:\n",
    "$$\n",
    "T = A*exp(B*tiempo)\n",
    "$$\n",
    "Realice una regresión lineal adecuadamente para predecir cuanto debería valer la temperatura si estamos en el tiempo 13.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<strong>(Intercept):</strong> 1.29733017399686"
      ],
      "text/latex": [
       "\\textbf{(Intercept):} 1.29733017399686"
      ],
      "text/markdown": [
       "**(Intercept):** 1.29733017399686"
      ],
      "text/plain": [
       "(Intercept) \n",
       "    1.29733 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# T = Aexp(Bt)\n",
    "# Log(T) = Log(A) + Bt\n",
    "log_T = log(T)\n",
    "reg4 = lm(log_T ~ tiempo)\n",
    "prediccion = exp(reg4$coef[1])*exp(reg4$coef[2]*13.5)\n",
    "prediccion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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BBASBBASBBASBBASBBASBBASBBASBBA\nSBBASBBASBBASIxibpfkCYkot/++0sQJiRgn7czvaUtCIkbXb9yf0eMoJEKctCOkkW5S4BDE\nElKKmxQ4BLFO23GMNM5NChyCQJv1upN2rNqNc5MChyDMvprTdpxHGuMmBQ5BmMO2aG7tnBAS\nT5rhykIHIfEkIW0IiScJaUNIPGt+a90dhMSz5rfW3UFIPG/W63U7QoIAQuIJNkUHQuJhDo6O\nhMTDLNcdCYlHOYHUIiQeJaQWIfEoIbUIiYc5RjoSEg+zanckJJ7gPNKBkCCAkCCAkCCAkCCA\nkCCAkCCAkBjTbNbHhcR4ZnTGVkiMZ0bXEAmJ0czpqlYhMRohRd+kwCEYn5Cib1LgEDyn13Kc\nY6TgmxQ4BM/ouRxn1S74JgUOwTN6b2qa1h9PmjQhMdywg59b26V7iVWToJAYbmBIVz/33q5f\nRbuGQmK4QSF1f/J2U3NvD7GixQoh8YAh3+FdIZ1uaq59nZqWz4XEA4bsc3WGdPUj925ZKiHx\nkAGrAJebr+bwNdYXH7q8YR3fDkJibJebr9NCHCONpoq7jt7ON19/IVm1G1cVdx3d+uz0/W1q\nnEcaVSV3Hpf6bUQq2tT0JCRC9T2sqWZT05OQiFTTQlsoIRFJSKPepMAhGIWQRr1JgUMwjopO\n/YQSEqGmtx7Xj5AI9ux6XJ3reUIimV6naivdogmJRHqeqm29rImQSKRXItWu+gmJNPolIqQe\nVu+L35cfL02z/DfSEBTreMX3rd27KyGVvwCRMKSfxe+9sVo0W8tRhqBc7Sfz3XrmRPuTD++r\nYAEiYUhvzevq98Xbz29Tb837GENQsObiZedndURTwwJEwpCaZrV/8buX1yzGGIKCnSRya+/u\nfNtTxXFT0pB+XyzaW/jwISha6/BoyCMspFNvzfd6/bF5sdki3TxIKvou4wlCeuomW9/N4v17\n/br4LenrpfkaYwiK98jxjmOkU1/7FbuNj3GGoHSPrMBZtTv37+1lU9Hrx0/Hl217fAhK98jD\nW/63hCsbIICQIECOkHr8pbdnh4C0hAQBhAQBhAQBhEQxyl/kvk5IFOL+adeSQ7P8TSHOLgS6\nvAi86OsbhEQZTi9N7aim7CvuhEQZzkI6ed/lx4sjJMpwEkpHNUJ6QKF3FmPq2hQJ6TmF3lmM\nqX1Y1FWNY6Qih6A8rYW6jmqs2hU5BEXrrMZ5pAKHoHAlV9NBSBBASORX2dani5DIrexVhJ6E\nRG5x69oZt2xCIrOwM61Zt2xCIrPBIV3b7mQ9YyskMhsY0mG7U9jv2hcSuQ3bktx5moWQEg9B\nMQYd2zSt26yFlH8ICjJgte2w+eq6GO/yXekIiarcCsmqXYYhqNRun65pvdH6mPNIyYegUl1P\nWyrgGiMhUZtmu/i9bm2XCrjGSEjU6fazAJMTErU7X/fOsp8nJGp3GlKm/TwhUbuzkE7el3wS\n496kwCGYjq6CUn8LCYnq3ftFXknmkOQmBQ7BlBzXF4SUeAgmyjFS2iGYKKt2aYdgspxHSjkE\nRBISBBASBBASBBASVcv/TKQdIVGxEp6JtCMkKlbCM5F2hES9cl0P1EFI1EtI+YdgAoSUfwim\nwDFS9iGYAqt22YdgGpxHyjwERBISczPKRkxIzMtIh1VCYl5GWugTErMy1qknITErQoIAQoII\njpEggFU7COE8EpRKSMzHiBfmCYm5GPVScSExF6M+eUlIzMD2D6EfXh9nhCQ3KXAIJq79V89P\n9umEBH2dtHO6Tyck6Ku5eOkYCYY6aefvDat2MEh3SM4jwSBde3PjflMJiSk6aSfFL+0SElN0\n1s74v7RLSExT4l94JyQIICQIICQ4eGJ3UEiw89TinpBg56nTTUJi1pqLq8If++YTEjPW3psT\nEjzo+kXij32lsW9S4BAQeUmekJivs5Cs2sEVN08NXawzOI8EHe5tZOKeYCEkJuxeKH+hPX2J\nq5CYrh4Lcc0uo/WzT1gSEtPVd0U7YA9PSExXz5CeO4P08G2FRC36bWqEBDf1O/gREtzRaznO\nMRIEsGoHIZxHghIICQIICQKkDOnnrVl8rNefL83ifaQhII+EIa0Wv0d0zefH5mWzHGUIyCRh\nSO/N73bofdG8rdar7evxQ0AmCUNabG/YNKvtP4sxhoDeYn85eMKQmub48s7ZLyExslvnYB9J\nLMMWafNyZYtEVtevCnrsMocMx0jvq/3r8UNAPzeuU33swjurdszR9ZAevBS8nPNITdujQ0Av\nVYdU1BDM29UdOCFBf9eXFIo/Rjp+gbtfQUiM7toRRPGrdscvICRKVvh5pOMXEBJTIyQIICTo\nNGwHT0jQYeiSg+Vv6DB0EVxIcGnwaVkhwSUhQQAhQQTHSBDAqh2EKP88UhFDQCQhQQAhQQAh\nQQAhwX13Vx6EBPf0WAsXEtzT4+yskOCOv+uFbuzgCQnu2H873tzBExLcctwM7f8KxJVPe+Ar\nPzqlooaAHlpbob+euj/xga/96KSKGgJ62C8ztHfqhAQDtZcZbj9FSUhw1Uk8NxfBhQRXnYZk\n1Q4ec7oVch4JHtL7ibJCglt6PlFWSBBASBBASBBASBBASBBASBBASBBASBBASBBASBBASBBA\nSBBASBBASBBASBBASBBASBBASBCg0JCgMg98l8eHU+yoJU0g/wxMIHQCQprrDExASBOYQP4Z\nmICQJjCB/DMwASFNYAL5Z2ACQprABPLPwASENIEJ5J+BCQhpAhPIPwMTENIEJpB/BiYgpAlM\nIP8MTEBIE5hA/hmYgJAmMIH8MzCBCYQEEyMkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAk\nCCAkCCAkCCAkCCAkCCAkCJAhpPdFs3hfpR934/Plb+x80/hvf6fnmcH3W9O8/eSbwKo1avoJ\nfB6+4aNnkT6k5fbX/b8kH3fjfTv2YpV1GqvF7k7PM4OvzHfBz2I3gZ8sE/g+/KGJ1tAxs0ge\n0n/N4nv9vWj+Sz3wenM3vq02P5Tesk7jdfdgZprB4nfU1WvznmsCb5uhf3+iZXkMfgfbfcO3\nhg6aRfKQ3puv35f/mo/UA68338LbfzZ3Zr5p/Nv/+Z08M/i3/T5eNYtcE2gyPgafzXI/fGvo\noFkkD+m12WzUv5vX1AMfbe7MbNP4OTyYeWbw1nwfXs0zgf1+7abk5BP4/SGyD6k1dNAskofU\n+omUyapZZpzGsvnZDZpnBi/N+mOx3cPNNIGP/a7dR4YJfJ+PufknaBYzDOlzsy3PNY2P5t86\nZ0hN87o91s82gfXnZrVh8ZlpAkIK87N4zTeN7R5E3pA2iw1vWTYIOx/bRbKPtZCeHDBzSKvF\nMuM0XjbrznlD2hwj/WxWe/NM4HOza/db8qeQnrPIHNLyJeM03rYrRLtB88yg9W2TZwIvzeb4\nbLUpOccE9oMtwu+GTKt2P5lW7X5elj8Zp9H++/N5ZtA6A5DrLsg6gZNVu5/jqt3Ts0ge0sf2\nh/LXdu0mua9mmXUa7ZDyzGA36s/mfsgzgd0GYHsiK8cE9iG1hg6axayubPj56yjrNHJe2fB7\ndLTaHKL8yzWB92ZzWdt7rksrpnNlw+9O8sby/ifGeztuD3JOY/9g5pnBx3HUPBNYZp3A4VDo\nJXoW6UPaXf2bfNiN1o5V3mls/8k0g6/lYdRMEziOmmECh5BW0bPId14UJkRIEEBIEEBIEEBI\nEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBI\nEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIEEBIZdv8oeCn/3Q94/MQFe1l+/gIqXwe\noqJJqBYeqKIJqRYeqJLt/wT7LqfPl2bxuXvv+qNZfKzX703zvnv7/e/vcv9+1stntgnPl5BK\n1g7pdfvGcvvej82rX8vNy/e/tzcfWi//PoukhFS05m+x4atZrtarZfO1efP31c/9y8Xm7cX3\n+nvR/Fuv/x1fJSkhFe0Y0muz+n1t1bxu3vxv+86f/YeaTV2/pb1uPmv3qk1SakIq2jGk5qD9\nzsOH1h2vkpR7vGhCqoV7vGgXtZy+U0jFcI8XrX2M9HX5zkM9m2Omr+bteIz0mmOysyakoh1X\nFLbrcevP3WLDen0a0m6p7suqXT5CKtpLs1vfXh/OEC1+ukLafmy7FXIeKRMhFe2/l7+QNtcs\nNG+HDdTZMdLr3+UMnwtXNuQgpPpZWiiAx6B+QiqAx6B+QiqAx6B+QiqAxwACCAkCCAkCCAkC\nCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkCCAkC\nCAkCCAkCCAkCCAkCCAkC/A+GcvjhAi6yGwAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(tiempo,log_T)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}