attachment_DSS_Stat_Inference_Hypothesis_Testing.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Statistical Inference and Hypothesis Testing\n",
    "\n",
    "**Statistics** is a science of inference. It is the science of generalization from a part(the randomly chosen sample) to the whole (the population).\n",
    "\n",
    "**Statistical inference** is the process of deducing properties of an underlying distribution by analysis of data. Inferential statistical analysis infers properties about a population: this includes testing hypotheses and deriving estimates.\n",
    "\n",
    "Statistics are helpful in analyzing most collections of data. **Hypothesis testing** can justify conclusions even when no scientific theory exists."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Average Experience of Data Science Specialization(DSS) batch with Statistical Inference"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "# Let us explore the original data\n",
    "\n",
    "dss_exp = np.array([12,  15,  13,  20,  19,  20,  11,  19,  11,  12,  19,  13,  \n",
    "                    12,  10,  6,  19,  3,  1,  1,  0,  4,  4,  6,  5,  3,  7,  \n",
    "                    12,  7,  9,  8,  12,  11,  11,  18,  19,  18,  19,  3,  6,  \n",
    "                     5,  6,  9,  11,  10,  14,  14,  16,  17,  17,  19,  0,  2,  \n",
    "                     0,  3,  1,  4,  6,  6,  8,  7,  7,  6,  7,  11,  11,  10,  \n",
    "                    11,  10,  13,  13,  15,  18,  20,  19,  1,  10,  8,  16,  \n",
    "                    19,  19,  17,  16,  11,  1,  10,  13,  15,  3,  8,  6,  9,  \n",
    "                    10,  15,  19,  2,  4,  5,  6,  9,  11,  10,  9,  10,  9,  \n",
    "                    15,  16,  18,  13])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Understanding the Underlying distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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ZmeFkYGZmOBmYmRlOBmZmhpOBmbVZX980JDX16uubVnbYo47vWmq9wXct7VqSSM+ha2qo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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2b7acaefdd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean Experience of DSS Batch: 10.435\n",
      "Std Deviation of Experience of DSS Batch: 5.665\n"
     ]
    }
   ],
   "source": [
    "# Plot the distribution of Experience\n",
    "plt.hist(dss_exp, range = (0,20), bins = 21)\n",
    "# Add axis labels\n",
    "plt.xlabel(\"Experience in years\")\n",
    "plt.ylabel(\"Frequency\")\n",
    "plt.title(\"Distribution of Experience in Data Science Specialization\")\n",
    "plt.axvline(x=dss_exp.mean(), linewidth=2, color = 'r')\n",
    "plt.show()\n",
    "\n",
    "# Statistics of DSS Batch experience\n",
    "print(\"Mean Experience of DSS Batch: {:4.3f}\".format(dss_exp.mean()))\n",
    "print(\"Std Deviation of Experience of DSS Batch: {:4.3f}\".format(dss_exp.std()))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Set the parameters for sampling\n",
    "n = 100\n",
    "NUM_TRIALS = 1000"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Estimating DSS Experience from samples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Samp_mean = 10.560 Sample_SD = 5.906\n"
     ]
    }
   ],
   "source": [
    "#Just try for 1 iteration\n",
    "\n",
    "samp = np.random.choice(dss_exp, size = n, replace = True)\n",
    "samp_mean = samp.mean()\n",
    "samp_sd = samp.std()\n",
    "print(\"Samp_mean = {:4.3f} Sample_SD = {:4.3f}\".format(samp_mean, samp_sd))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Distribution of Sample means\n",
    "\n",
    "mn_array = np.zeros(NUM_TRIALS)\n",
    "sd_array = np.zeros(NUM_TRIALS)\n",
    "\n",
    "# Extract Random Samples and compute mean & standard deviation\n",
    "for i in range(NUM_TRIALS):\n",
    "  samp = np.random.choice(dss_exp, size = n, replace = True)\n",
    "  mn_array[i] = samp.mean()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean = 10.446, Std Dev = 0.566, 5% Pct = 9.559, 95% Pct = 11.391\n"
     ]
    },
    {
     "data": {
      "image/png": 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akkZHxOPNXMdQRMTHIuJ5YGvg5OrJTV79XGo/uuEzwNMRsQtwDnAmpPc2AF/P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\nSZiZWSknCTMzK/X/AaUnOV61qAX4AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2b7acd4a080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Annotate the distribution - Not important for understanding\n",
    "mn = mn_array.mean()\n",
    "sd = mn_array.std()\n",
    "x5_pct = np.percentile(mn_array, 5.0)\n",
    "x95_pct = np.percentile(mn_array, 95.0)\n",
    "print(\"Mean = {:4.3f}, Std Dev = {:4.3f}, 5% Pct = {:4.3f}, 95% Pct = {:4.3f}\".format(mn, sd, x5_pct, x95_pct))\n",
    "\n",
    "\n",
    "# Plot Sampling distribution of Mean    \n",
    "plt.hist(mn_array, range=(0,20), bins = 41)\n",
    "# Add axis labels\n",
    "plt.xlabel(\"Avg Experience with n={}\".format(n))\n",
    "plt.ylabel(\"Frequency\")\n",
    "plt.title(\"Sampling Distribution of Mean\")\n",
    "plt.axvline(x=x5_pct, linewidth=2, color = 'r')\n",
    "plt.axvline(x=x95_pct, linewidth=2, color = 'r')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Function to check if the true mean lies within 90% Confidence Interval\n",
    "\n",
    "def samp_mean_within_ci(mn, l_5pct, u_95pct):\n",
    "    out = True\n",
    "    if (mn < l_5pct) | (mn > u_95pct):\n",
    "        out = False\n",
    "        \n",
    "    return out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample: L_5PCT = 9.801 M = samp_mean = 10.740 H_95PCT = 11.679\n",
      "True mean lies with the 90% confidence Intervel = True\n"
     ]
    }
   ],
   "source": [
    "# Estimation and Confidence Interval\n",
    "#np.random.seed(100) # seed value is set to reproduce the results\n",
    "\n",
    "samp = np.random.choice(dss_exp, size = n, replace = True)\n",
    "samp_mean = samp.mean()\n",
    "samp_sd = samp.std()\n",
    "sd_ci = samp_sd/np.sqrt(n)\n",
    "\n",
    "samp_lower_5pct = samp_mean - 1.645 * sd_ci\n",
    "samp_upper_95pct = samp_mean + 1.645 * sd_ci\n",
    "print(\"Sample: L_5PCT = {:4.3f} M = samp_mean = {:4.3f} H_95PCT = {:4.3f}\".format(samp_lower_5pct, samp_mean, samp_upper_95pct))\n",
    "mn_within_ci_flag = samp_mean_within_ci(samp_mean, x5_pct, x95_pct)\n",
    "print(\"True mean lies with the 90% confidence Intervel = {}\".format(mn_within_ci_flag))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hypothesis Testing\n",
    "\n",
    "Let us define the Hypotheses as follows:\n",
    "- **H0** : Average Experience of Current Batch & Previous batch are **same**\n",
    "- **H1** : Average Experience of Current Batch & Previous batch are **different**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Previous Batch Data for working experience\n",
    "dss_exp_prev = np.array([1, 14,  6,  7, 10, 10, 19, 15, 19, 15, \n",
    "                    2,  2, 14, 14, 14,  3,  0,  4, 11,  7, \n",
    "                    1,  2,  0,  1,  2,  2,  2,  1,  1,  2, \n",
    "                    4,  4,  3,  3,  3,  3,  4,  3,  3,  7, \n",
    "                    8,  6,  6,  6,  7, 8, 8, 8, 8, 7, \n",
    "                    8, 0, 0, 7, 6, 9, 10, 9, 9, 11, \n",
    "                    11, 9, 10, 10, 11, 10, 11, 9, 9, 9, \n",
    "                    12, 14, 13, 14, 18, 14, 11, 10, 17, 20, \n",
    "                    18, 5, 13, 4, 2, 4, 3, 12, 12, 14, \n",
    "                    12, 12, 10, 14, 4, 11, 9])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "avg_exp_prev = dss_exp_prev.mean()\n",
    "std_exp_prev = dss_exp_prev.std()\n",
    "print(\"Previous DSS Batch: Avg Exp - {:4.3f} Std Dev - {:4.3f}\".format(avg_exp_prev, std_exp_prev))\n",
    "\n",
    "plt.hist(dss_exp_prev, range=(0,20), bins = 21)\n",
    "plt.axvline(x=dss_exp_prev.mean(), linewidth=2, color = 'r')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "np.random.seed(100)\n",
    "n = 20\n",
    "\n",
    "dss_mean = dss_exp.mean()\n",
    "dss_sd   = dss_exp.std()\n",
    "print(\"Current DSS Batch : Population Mean - {:4.3f}\".format(dss_mean))\n",
    "\n",
    "dss_prev_samp = np.random.choice(dss_exp_prev, size = n, replace = True)\n",
    "dss_prev_samp_mean = dss_prev_samp.mean()\n",
    "print(\"Previous DSS Batch Sample Mean: {:4.3f}\".format(dss_prev_samp_mean))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy import stats\n",
    "t_statistic = (dss_prev_samp_mean - dss_mean)/(dss_sd/np.sqrt(n))\n",
    "p_val = 2 * stats.t.cdf(t_statistic, df= (n-1))\n",
    "print(\"T-Statistic : {:4.2f}, p-Value = {:4.2f}\".format(t_statistic,p_val))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Summary\n",
    "Above code describes to do hypothesis testing based on the intuition from Sampling distribution of mean. There will be differences with exact math based on various conditions and assumptions made."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Appendix Code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from collections import Counter\n",
    "c = Counter(dss_exp)\n",
    "print(c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Create random samples to check if the true mean lie within the specified confidence Interval\n",
    "\n",
    "samp_mn_vec, lower_5pct, upper_95pct, flag_true_mean_within_ci = (np.zeros(NUM_TRIALS), np.zeros(NUM_TRIALS), np.zeros(NUM_TRIALS), np.zeros(NUM_TRIALS))\n",
    "for i in range(NUM_TRIALS):\n",
    "    samp = np.random.choice(dss_exp, size = n, replace = True)\n",
    "    samp_mn_vec[i] = samp.mean()\n",
    "    samp_sd = samp.std()\n",
    "    sd_ci = samp_sd/np.sqrt(n)\n",
    "    lower_5pct[i] = samp_mn_vec[i] - 1.645 * sd_ci\n",
    "    upper_95pct[i] = samp_mn_vec[i] + 1.645 * sd_ci\n",
    "    flag_true_mean_within_ci[i] = int(samp_mean_within_ci(samp_mn_vec[i], x5_pct, x95_pct))\n",
    "    #print(x5_pct, samp_mn_vec[i], x95_pct, lower_5pct[i], upper_95pct[i], flag_true_mean_within_ci[i])\n",
    "    \n",
    "df = pd.DataFrame(np.column_stack((samp_mn_vec,lower_5pct,upper_95pct, flag_true_mean_within_ci)), columns = ['samp_mean', 'lower_5pct', 'upper_95pct', 'flag_true_mean_within_ci' ])    \n",
    "df.to_csv('expt_n{}_iters{}.csv'.format(n, NUM_TRIALS)) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# For 2-tailed hypothesis testing \n",
    "# Ref: http://www.scipy-lectures.org/packages/statistics/index.html#hypothesis-testing-comparing-two-groups\n",
    "from scipy import stats\n",
    "dss_exp_prev_samp = np.random.choice(dss_exp_prev, size = 20, replace = True)\n",
    "dss_exp_samp = np.random.choice(dss_exp, size = 20, replace = True)\n",
    "stats.ttest_ind(dss_exp_prev_samp, dss_exp_samp)"
   ]
  }
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