{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Illustration of Confidence Intervals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example 4.1. Illustration of Confidence Intervals.** \n",
    "Assume a sample of N= 31 independent observations are collected \n",
    "from a normally distributed random variable x with the following results:\n",
    "\n",
    "    60, 61, 47, 56, 61, 63, \n",
    "    65, 69, 54, 59, 43, 61, \n",
    "    55, 61, 56, 48, 67, 65, \n",
    "    60, 58, 57, 62, 57, 58, \n",
    "    53, 59, 58, 61, 67, 62, 54\n",
    "\n",
    "Determine a 90% confidence interval for the mean value and variance of the random variable x."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solution\n",
    "Let's go to chapter 4 (Statistical Principals), section 4 (**Confidence Intervals**) of J. Bendat, A Piersol."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"100%\"\n",
       "            height=\"400\"\n",
       "            src=\"pdfs/J-Bendat--A-Piersol-random-data-analysis-and-measurement-procedures-section-4-4.pdf\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.IFrame at 0x110641b10>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import IFrame\n",
    "IFrame('pdfs/J-Bendat--A-Piersol-random-data-analysis-and-measurement-procedures-section-4-4.pdf',\n",
    "       width='100%', height=400)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x108bfa850>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x108bee790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "import seaborn as sns\n",
    "%matplotlib inline\n",
    "\n",
    "# The data\n",
    "X = np.array([60, 61, 47, 56, 61, 63, 65, 69, 54, 59, 43, 61, 55, 61,\n",
    "              56, 48, 67, 65, 60, 58, 57, 62, 57, 58, 53, 59, 58, 61, 67, 62, 54])\n",
    "\n",
    "# Let's visualize the distribution\n",
    "sns.distplot(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean = 58.61\n",
      "s2 = 33.45\n",
      "s = 5.78\n",
      "1.69726089436\n",
      "confidence interval for μ_x: (56.85 < 58.61 < 60.38)\n",
      "confidence interval for μ_x: (56.82, 60.40)\n"
     ]
    }
   ],
   "source": [
    "# Number of data\n",
    "N = float(len(X))\n",
    "\n",
    "# We estimate the sample mean\n",
    "mu = X.mean()\n",
    "\n",
    "# We estimate the sample variance and standar deviation with N-1 degrees of freedom\n",
    "s2 = X.var(ddof=1)\n",
    "s = X.std(ddof=1)\n",
    "\n",
    "# We print the values\n",
    "print 'mean = %4.2f' % mu\n",
    "print 's2 = %4.2f' % s2\n",
    "print 's = %4.2f' % s\n",
    "\n",
    "\n",
    "# We need to determine confidence intervals with 90% confidence\n",
    "# 90% confidence means an alpha = 0.1\n",
    "# The confidence intervals for μ_x are given by Eq. (4.46b)\n",
    "\n",
    "# We need the t-student statistic for t(1-alpha/2,N-1)\n",
    "t = stats.t.ppf(1-0.1/2,N-1) # ppf: percent point function (inverse of cdf)\n",
    "\n",
    "# The confidence interval\n",
    "conf_int_mu = (mu-t*s/np.sqrt(N),mu+t*s/np.sqrt(N))\n",
    "\n",
    "print 'confidence interval for μ_x: (%2.2f < %2.2f < %2.2f)' % (conf_int_mu[0],mu,conf_int_mu[1])\n",
    "# Another way to do it using stats.t.interval \n",
    "print 'confidence interval for μ_x: (%2.2f, %2.2f)' % stats.t.interval(1-0.1, N-1, loc=mu, scale=s/np.sqrt(N-1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Additional work \n",
    "1. Compute the 90% confidence interval for the variance $s^2_x$. Use Eq. (4.47) and the function [``stats.chi2.ppf()``.](http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.chi2.html#scipy.stats.chi2 \"scipy.stats.chi2 — SciPy v0.17.0 Reference Guide\")\n",
    "2. Compute the 95% confidence intervals for the mean $\\mu_x$ and variance $s^2_x$."
   ]
  }
 ],
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