AB_Testing.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analyze A/B Test Results\n",
    "\n",
    "## Table of Contents\n",
    "- [Introduction](#intro)\n",
    "- [Part I - Probability](#probability)\n",
    "- [Part II - A/B Test](#ab_test)\n",
    "- [Part III - Regression](#regression)\n",
    "\n",
    "<a id='intro'></a>\n",
    "### Introduction\n",
    "\n",
    "A/B tests are very commonly performed by data analysts and data scientists. For this project, you will be working to understand the results of an A/B test run by an e-commerce website.  Your goal is to work through this notebook to help the company understand if they should implement the new page, keep the old page, or perhaps run the experiment longer to make their decision.\n",
    "\n",
    "<a id='probability'></a>\n",
    "#### Part I - Probability"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import random\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "random.seed(42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`1.` Now, read in the `ab_data.csv` data. Store it in `df`.  **Use your dataframe to answer the questions in Quiz 1 of the classroom.**\n",
    "\n",
    "a. Read in the dataset and take a look at the top few rows here:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>851104</td>\n",
       "      <td>2017-01-21 22:11:48.556739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>804228</td>\n",
       "      <td>2017-01-12 08:01:45.159739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>661590</td>\n",
       "      <td>2017-01-11 16:55:06.154213</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>853541</td>\n",
       "      <td>2017-01-08 18:28:03.143765</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>864975</td>\n",
       "      <td>2017-01-21 01:52:26.210827</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted\n",
       "0   851104  2017-01-21 22:11:48.556739    control     old_page          0\n",
       "1   804228  2017-01-12 08:01:45.159739    control     old_page          0\n",
       "2   661590  2017-01-11 16:55:06.154213  treatment     new_page          0\n",
       "3   853541  2017-01-08 18:28:03.143765  treatment     new_page          0\n",
       "4   864975  2017-01-21 01:52:26.210827    control     old_page          1"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv('ab_data.csv')\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Use the cell below to find the number of rows in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(294478, 5)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. The number of unique users in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290584"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# df.user_id.unique()\n",
    "df.user_id.nunique()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "d. The proportion of users converted."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11965919355605512"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# df.query('converted == 1').user_id.mean()\n",
    "df.converted.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "e. The number of times the `new_page` and `treatment` don't match."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3893"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.query(\"group == 'treatment' and not landing_page == 'new_page' or group == 'control' and landing_page == 'new_page'\").count()[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Do any of the rows have missing values?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "user_id         0\n",
       "timestamp       0\n",
       "group           0\n",
       "landing_page    0\n",
       "converted       0\n",
       "dtype: int64"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.isnull().sum() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`2.` For the rows where **treatment** does not match with **new_page** or **control** does not match with **old_page**, we cannot be sure if this row truly received the new or old page.  Use **Quiz 2** in the classroom to figure out how we should handle these rows.  \n",
    "\n",
    "a. Now use the answer to the quiz to create a new dataset that meets the specifications from the quiz.  Store your new dataframe in **df2**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "df2 = df.copy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "no_matches = df2[((df2['group'] == 'treatment') == (df2['landing_page'] == 'new_page')) == False].index"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "df2.drop(no_matches, inplace=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Double Check all of the correct rows were removed - this should be 0\n",
    "df2[((df2['group'] == 'treatment') == (df2['landing_page'] == 'new_page')) == False].shape[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
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       "      <td>851104</td>\n",
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       "      <td>control</td>\n",
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       "      <th>1</th>\n",
       "      <td>804228</td>\n",
       "      <td>2017-01-12 08:01:45.159739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>661590</td>\n",
       "      <td>2017-01-11 16:55:06.154213</td>\n",
       "      <td>treatment</td>\n",
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       "      <th>3</th>\n",
       "      <td>853541</td>\n",
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       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>864975</td>\n",
       "      <td>2017-01-21 01:52:26.210827</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted\n",
       "0   851104  2017-01-21 22:11:48.556739    control     old_page          0\n",
       "1   804228  2017-01-12 08:01:45.159739    control     old_page          0\n",
       "2   661590  2017-01-11 16:55:06.154213  treatment     new_page          0\n",
       "3   853541  2017-01-08 18:28:03.143765  treatment     new_page          0\n",
       "4   864975  2017-01-21 01:52:26.210827    control     old_page          1"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`3.` Use **df2** and the cells below to answer questions for **Quiz3** in the classroom."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a. How many unique **user_id**s are in **df2**?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290584"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.user_id.nunique()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "b. There is one **user_id** repeated in **df2**.  What is it?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
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       "      <th>2893</th>\n",
       "      <td>773192</td>\n",
       "      <td>2017-01-14 02:55:59.590927</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "      user_id                   timestamp      group landing_page  converted\n",
       "2893   773192  2017-01-14 02:55:59.590927  treatment     new_page          0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2.user_id.duplicated()]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is the row information for the repeat **user_id**? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
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       "      <td>2017-01-14 02:55:59.590927</td>\n",
       "      <td>treatment</td>\n",
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       "      <td>0</td>\n",
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      "text/plain": [
       "      user_id                   timestamp      group landing_page  converted\n",
       "2893   773192  2017-01-14 02:55:59.590927  treatment     new_page          0"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2.user_id.duplicated()]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. Remove **one** of the rows with a duplicate **user_id**, but keep your dataframe as **df2**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "duplicate_user = df2[df2.user_id.duplicated()].index # save duplicated row index to variable name "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "df2.drop(duplicate_user, inplace=True) # drop duplicated user_id row "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "Empty DataFrame\n",
       "Columns: [user_id, timestamp, group, landing_page, converted]\n",
       "Index: []"
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     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "df2[df2.user_id.duplicated()] # check to confirm duplicate removed"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`4.` Use **df2** in the cells below to answer the quiz questions related to **Quiz 4** in the classroom.\n",
    "\n",
    "a. What is the probability of an individual converting regardless of the page they receive?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11.959708724499627"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.query(\"converted == 1\").count()[0]/ df2.converted.shape[0]*100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Given that an individual was in the `control` group, what is the probability they converted?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12.03863045004612"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.query(\"group == 'control' and converted == 1\").count()[0] /df2.query(\"group == 'control'\").count()[0]*100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Given that an individual was in the `treatment` group, what is the probability they converted?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11.880806551510565"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.query(\"group == 'treatment' and converted == 1\").count()[0]/ df2.query(\"group == 'treatment'\").count()[0]*100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is the probability that an individual received the new page?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50.00619442226688"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.query(\"landing_page == 'new_page'\").count()[0] / df2.landing_page.shape[0]*100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11.959708724499627"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2['converted']==1].count()[0] / df2.converted.shape[0]*100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "88.04029127550037"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2['converted']==0].count()[0] / df2.converted.shape[0]*100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.15782389853555567"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "treatment = df2[df2['group'] == 'treatment']['converted'].mean()\n",
    "control =  df2[df2['group'] == 'control']['converted'].mean()\n",
    "obs_diff = control - treatment # observed differnces in the means\n",
    "obs_diff*100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('2017-01-24 13:41:54.460509', '2017-01-02 13:42:05.378582')"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import datetime\n",
    "df2.timestamp.max(), df.timestamp.min() # getting min, max of timestamps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<pandas.core.groupby.groupby.DataFrameGroupBy object at 0x114259d68>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.timestamp = pd.to_datetime(df2['timestamp']) # group by sort timestamp \n",
    "df2.timestamp.astype(str) \n",
    "df2.groupby('timestamp')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "scrolled": true
   },
   "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>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>193652</th>\n",
       "      <td>920411</td>\n",
       "      <td>2017-01-24 13:41:54.460509</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>179072</th>\n",
       "      <td>836373</td>\n",
       "      <td>2017-01-24 13:41:52.604673</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>157787</th>\n",
       "      <td>843121</td>\n",
       "      <td>2017-01-24 13:41:44.097174</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>47535</th>\n",
       "      <td>808330</td>\n",
       "      <td>2017-01-24 13:41:19.152664</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>158814</th>\n",
       "      <td>799244</td>\n",
       "      <td>2017-01-24 13:41:15.325359</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>180924</th>\n",
       "      <td>808180</td>\n",
       "      <td>2017-01-24 13:41:07.867394</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29690</th>\n",
       "      <td>787704</td>\n",
       "      <td>2017-01-24 13:41:04.416938</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>260161</th>\n",
       "      <td>712000</td>\n",
       "      <td>2017-01-24 13:40:54.479759</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>178372</th>\n",
       "      <td>891992</td>\n",
       "      <td>2017-01-24 13:40:37.103733</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24692</th>\n",
       "      <td>725007</td>\n",
       "      <td>2017-01-24 13:40:23.844184</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>225633</th>\n",
       "      <td>735831</td>\n",
       "      <td>2017-01-24 13:40:20.250510</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29756</th>\n",
       "      <td>770192</td>\n",
       "      <td>2017-01-24 13:40:16.549655</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>156537</th>\n",
       "      <td>745849</td>\n",
       "      <td>2017-01-24 13:40:11.453255</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>260331</th>\n",
       "      <td>697239</td>\n",
       "      <td>2017-01-24 13:39:54.628956</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>181845</th>\n",
       "      <td>941363</td>\n",
       "      <td>2017-01-24 13:39:47.646844</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>55486</th>\n",
       "      <td>781406</td>\n",
       "      <td>2017-01-24 13:39:34.441847</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>223456</th>\n",
       "      <td>636883</td>\n",
       "      <td>2017-01-24 13:39:25.679382</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>251684</th>\n",
       "      <td>648387</td>\n",
       "      <td>2017-01-24 13:39:25.446588</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>72311</th>\n",
       "      <td>693559</td>\n",
       "      <td>2017-01-24 13:39:22.263167</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>248023</th>\n",
       "      <td>806188</td>\n",
       "      <td>2017-01-24 13:39:18.170631</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>189279</th>\n",
       "      <td>705642</td>\n",
       "      <td>2017-01-24 13:39:17.861197</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>181369</th>\n",
       "      <td>722521</td>\n",
       "      <td>2017-01-24 13:39:15.092095</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11698</th>\n",
       "      <td>868142</td>\n",
       "      <td>2017-01-24 13:39:06.382676</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>221628</th>\n",
       "      <td>786852</td>\n",
       "      <td>2017-01-24 13:39:03.691748</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>270808</th>\n",
       "      <td>654334</td>\n",
       "      <td>2017-01-24 13:39:02.937594</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28671</th>\n",
       "      <td>849301</td>\n",
       "      <td>2017-01-24 13:39:01.922937</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20234</th>\n",
       "      <td>676307</td>\n",
       "      <td>2017-01-24 13:38:56.135529</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>91879</th>\n",
       "      <td>718903</td>\n",
       "      <td>2017-01-24 13:38:50.618660</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>60565</th>\n",
       "      <td>893948</td>\n",
       "      <td>2017-01-24 13:38:49.847407</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>172036</th>\n",
       "      <td>789684</td>\n",
       "      <td>2017-01-24 13:38:46.564903</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>259791</th>\n",
       "      <td>790010</td>\n",
       "      <td>2017-01-02 13:45:25.265864</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>178400</th>\n",
       "      <td>932084</td>\n",
       "      <td>2017-01-02 13:45:21.968916</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>239917</th>\n",
       "      <td>643348</td>\n",
       "      <td>2017-01-02 13:45:14.834730</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>270313</th>\n",
       "      <td>664064</td>\n",
       "      <td>2017-01-02 13:45:13.056529</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>203173</th>\n",
       "      <td>901354</td>\n",
       "      <td>2017-01-02 13:45:06.600508</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27089</th>\n",
       "      <td>752901</td>\n",
       "      <td>2017-01-02 13:45:00.214893</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>285088</th>\n",
       "      <td>671039</td>\n",
       "      <td>2017-01-02 13:44:43.808624</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>128503</th>\n",
       "      <td>889231</td>\n",
       "      <td>2017-01-02 13:44:38.722463</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>292648</th>\n",
       "      <td>939117</td>\n",
       "      <td>2017-01-02 13:44:29.563089</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>208007</th>\n",
       "      <td>738560</td>\n",
       "      <td>2017-01-02 13:44:27.385563</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>220012</th>\n",
       "      <td>758991</td>\n",
       "      <td>2017-01-02 13:44:15.657021</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>109564</th>\n",
       "      <td>839956</td>\n",
       "      <td>2017-01-02 13:43:59.531519</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>235862</th>\n",
       "      <td>842582</td>\n",
       "      <td>2017-01-02 13:43:52.915614</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>217465</th>\n",
       "      <td>644166</td>\n",
       "      <td>2017-01-02 13:43:52.169597</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3179</th>\n",
       "      <td>884319</td>\n",
       "      <td>2017-01-02 13:43:36.945202</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>294185</th>\n",
       "      <td>889153</td>\n",
       "      <td>2017-01-02 13:43:35.689998</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>189698</th>\n",
       "      <td>677659</td>\n",
       "      <td>2017-01-02 13:43:32.744957</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>52817</th>\n",
       "      <td>937131</td>\n",
       "      <td>2017-01-02 13:43:24.473268</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>234314</th>\n",
       "      <td>667072</td>\n",
       "      <td>2017-01-02 13:43:22.081120</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>126997</th>\n",
       "      <td>643921</td>\n",
       "      <td>2017-01-02 13:43:03.767329</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>112584</th>\n",
       "      <td>799109</td>\n",
       "      <td>2017-01-02 13:42:45.219901</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26725</th>\n",
       "      <td>781125</td>\n",
       "      <td>2017-01-02 13:42:45.090432</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1985</th>\n",
       "      <td>688937</td>\n",
       "      <td>2017-01-02 13:42:41.406986</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>111830</th>\n",
       "      <td>722516</td>\n",
       "      <td>2017-01-02 13:42:34.033708</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>191378</th>\n",
       "      <td>762651</td>\n",
       "      <td>2017-01-02 13:42:28.522322</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>158789</th>\n",
       "      <td>725857</td>\n",
       "      <td>2017-01-02 13:42:27.851110</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>102717</th>\n",
       "      <td>818377</td>\n",
       "      <td>2017-01-02 13:42:26.640581</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>83878</th>\n",
       "      <td>737319</td>\n",
       "      <td>2017-01-02 13:42:21.786186</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>184884</th>\n",
       "      <td>781507</td>\n",
       "      <td>2017-01-02 13:42:15.234051</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>131228</th>\n",
       "      <td>922696</td>\n",
       "      <td>2017-01-02 13:42:05.378582</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>290584 rows × 5 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "        user_id                  timestamp      group landing_page  converted\n",
       "193652   920411 2017-01-24 13:41:54.460509    control     old_page          0\n",
       "179072   836373 2017-01-24 13:41:52.604673    control     old_page          0\n",
       "157787   843121 2017-01-24 13:41:44.097174  treatment     new_page          0\n",
       "47535    808330 2017-01-24 13:41:19.152664    control     old_page          0\n",
       "158814   799244 2017-01-24 13:41:15.325359  treatment     new_page          0\n",
       "180924   808180 2017-01-24 13:41:07.867394    control     old_page          0\n",
       "29690    787704 2017-01-24 13:41:04.416938  treatment     new_page          0\n",
       "260161   712000 2017-01-24 13:40:54.479759    control     old_page          1\n",
       "178372   891992 2017-01-24 13:40:37.103733    control     old_page          0\n",
       "24692    725007 2017-01-24 13:40:23.844184    control     old_page          0\n",
       "225633   735831 2017-01-24 13:40:20.250510  treatment     new_page          0\n",
       "29756    770192 2017-01-24 13:40:16.549655    control     old_page          0\n",
       "156537   745849 2017-01-24 13:40:11.453255  treatment     new_page          0\n",
       "260331   697239 2017-01-24 13:39:54.628956  treatment     new_page          1\n",
       "181845   941363 2017-01-24 13:39:47.646844    control     old_page          0\n",
       "55486    781406 2017-01-24 13:39:34.441847    control     old_page          0\n",
       "223456   636883 2017-01-24 13:39:25.679382  treatment     new_page          0\n",
       "251684   648387 2017-01-24 13:39:25.446588    control     old_page          0\n",
       "72311    693559 2017-01-24 13:39:22.263167  treatment     new_page          0\n",
       "248023   806188 2017-01-24 13:39:18.170631  treatment     new_page          0\n",
       "189279   705642 2017-01-24 13:39:17.861197    control     old_page          1\n",
       "181369   722521 2017-01-24 13:39:15.092095  treatment     new_page          0\n",
       "11698    868142 2017-01-24 13:39:06.382676  treatment     new_page          0\n",
       "221628   786852 2017-01-24 13:39:03.691748    control     old_page          0\n",
       "270808   654334 2017-01-24 13:39:02.937594  treatment     new_page          0\n",
       "28671    849301 2017-01-24 13:39:01.922937    control     old_page          0\n",
       "20234    676307 2017-01-24 13:38:56.135529  treatment     new_page          0\n",
       "91879    718903 2017-01-24 13:38:50.618660  treatment     new_page          1\n",
       "60565    893948 2017-01-24 13:38:49.847407    control     old_page          0\n",
       "172036   789684 2017-01-24 13:38:46.564903  treatment     new_page          0\n",
       "...         ...                        ...        ...          ...        ...\n",
       "259791   790010 2017-01-02 13:45:25.265864    control     old_page          0\n",
       "178400   932084 2017-01-02 13:45:21.968916    control     old_page          0\n",
       "239917   643348 2017-01-02 13:45:14.834730    control     old_page          0\n",
       "270313   664064 2017-01-02 13:45:13.056529    control     old_page          1\n",
       "203173   901354 2017-01-02 13:45:06.600508    control     old_page          0\n",
       "27089    752901 2017-01-02 13:45:00.214893    control     old_page          0\n",
       "285088   671039 2017-01-02 13:44:43.808624  treatment     new_page          0\n",
       "128503   889231 2017-01-02 13:44:38.722463    control     old_page          0\n",
       "292648   939117 2017-01-02 13:44:29.563089    control     old_page          0\n",
       "208007   738560 2017-01-02 13:44:27.385563  treatment     new_page          0\n",
       "220012   758991 2017-01-02 13:44:15.657021  treatment     new_page          0\n",
       "109564   839956 2017-01-02 13:43:59.531519    control     old_page          0\n",
       "235862   842582 2017-01-02 13:43:52.915614    control     old_page          0\n",
       "217465   644166 2017-01-02 13:43:52.169597    control     old_page          0\n",
       "3179     884319 2017-01-02 13:43:36.945202  treatment     new_page          0\n",
       "294185   889153 2017-01-02 13:43:35.689998  treatment     new_page          1\n",
       "189698   677659 2017-01-02 13:43:32.744957    control     old_page          0\n",
       "52817    937131 2017-01-02 13:43:24.473268  treatment     new_page          0\n",
       "234314   667072 2017-01-02 13:43:22.081120  treatment     new_page          0\n",
       "126997   643921 2017-01-02 13:43:03.767329    control     old_page          1\n",
       "112584   799109 2017-01-02 13:42:45.219901    control     old_page          0\n",
       "26725    781125 2017-01-02 13:42:45.090432  treatment     new_page          0\n",
       "1985     688937 2017-01-02 13:42:41.406986  treatment     new_page          0\n",
       "111830   722516 2017-01-02 13:42:34.033708  treatment     new_page          0\n",
       "191378   762651 2017-01-02 13:42:28.522322  treatment     new_page          0\n",
       "158789   725857 2017-01-02 13:42:27.851110  treatment     new_page          0\n",
       "102717   818377 2017-01-02 13:42:26.640581  treatment     new_page          0\n",
       "83878    737319 2017-01-02 13:42:21.786186    control     old_page          0\n",
       "184884   781507 2017-01-02 13:42:15.234051    control     old_page          0\n",
       "131228   922696 2017-01-02 13:42:05.378582  treatment     new_page          0\n",
       "\n",
       "[290584 rows x 5 columns]"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.groupby('timestamp')\n",
    "df2.sort_values('timestamp', ascending=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "end = '2017-01-24'\n",
    "start = '2017-01-02'       "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "from datetime import datetime\n",
    "def days_between(d1, d2):\n",
    "    d1 = datetime.strptime(d1, \"%Y-%m-%d\")\n",
    "    d2 = datetime.strptime(d2, \"%Y-%m-%d\")\n",
    "    return abs((d2 - d1).days)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "22"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "days_between(start, end) # get duration of A/B test period"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Consider your results from parts (a) through (d) above, and explain below whether you think there is sufficient evidence to conclude that the new treatment page leads to more conversions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**ANSWER**:\n",
    "\n",
    "\n",
    "In this dataset which spanned over the course of `22` days, we had `290584` unique users, where the probability that they convert regardless of what page they recieved either, `landing_page` or `old_page`, was `11.95%`. If the user was in the `control` group the probability they'd convert was `12.03%` while if the user was in the `treatment` group their probability of conversion was `11.88%`. Finally the probability the user received the `new_page` was `50%`. \n",
    "\n",
    "Calculating the `observed difference` between the conversion rates we get about a `0.0015` difference which is minimal but, in order to deem `statistically significant` we would have to perform a `bootstrap` on the sample to \"simulate the sampling distributions for the difference in proportions\" (cite 1).\n",
    "\n",
    "With these findings it is difficult to say which page leads were more effective. \n",
    "\n",
    "As we learned in the `A/B Tests` lesson there are important questions to consider when concluding results from our statistics. We haven't yet introduced `Hypothesis Testing` in which we would introduce sampling, and other techniques to show more useful metrics.  \n",
    "\n",
    "In conclusion, the next best option is to continue with the testing but, for a longer duration to rule out `response bias` or `change aversion` and begin with `Hypothesis Testing` to look at new metrics like `p-values`, `confidence intervals`, etc. \n",
    "\n",
    "\n",
    "> Citations:\n",
    "1. https://classroom.udacity.com/nanodegrees/nd002/parts/bb0cbeb6-d3f8-4bae-9dc3-4abba3823e54/modules/82c7c576-88f5-43b6-80fc-c53378527a2d/lessons/f3088061-6562-445e-b1a7-a86c9389c2b5/concepts/0fc48912-c4a4-498a-9217-b44330857e52"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='ab_test'></a>\n",
    "### Part II - A/B Test\n",
    "\n",
    "Notice that because of the time stamp associated with each event, you could technically run a hypothesis test continuously as each observation was observed.  \n",
    "\n",
    "However, then the hard question is do you stop as soon as one page is considered significantly better than another or does it need to happen consistently for a certain amount of time?  How long do you run to render a decision that neither page is better than another?  \n",
    "\n",
    "These questions are the difficult parts associated with A/B tests in general.  \n",
    "\n",
    "\n",
    "`1.` For now, consider you need to make the decision just based on all the data provided.  If you want to assume that the old page is better unless the new page proves to be definitely better at a Type I error rate of 5%, what should your null and alternative hypotheses be?  You can state your hypothesis in terms of words or in terms of **$p_{old}$** and **$p_{new}$**, which are the converted rates for the old and new pages."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. **Null Hypothesis**: \n",
    "> $$H_0:   p_{old}  \\leq   p_{new} >     Type I Error Rate .05 $$\n",
    "2. **Alternative Hypothesis**: \n",
    "> $$H_1: p_{new}  \\geq  p_{old}  < Type I Error Rate of .05 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`2.` Assume under the null hypothesis, $p_{new}$ and $p_{old}$ both have \"true\" success rates equal to the **converted** success rate regardless of page - that is $p_{new}$ and $p_{old}$ are equal. Furthermore, assume they are equal to the **converted** rate in **ab_data.csv** regardless of the page. <br><br>\n",
    "\n",
    "Use a sample size for each page equal to the ones in **ab_data.csv**.  <br><br>\n",
    "\n",
    "Perform the sampling distribution for the difference in **converted** between the two pages over 10,000 iterations of calculating an estimate from the null.  <br><br>\n",
    "\n",
    "Use the cells below to provide the necessary parts of this simulation.  If this doesn't make complete sense right now, don't worry - you are going to work through the problems below to complete this problem.  You can use **Quiz 5** in the classroom to make sure you are on the right track.<br><br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. **Null Hypothesis**: \n",
    "> $$H_0:  \\mu_{old} - \\mu_{new} = 0$$\n",
    "2. **Alternative Hypothesis**: \n",
    "> $$H_1: \\mu_{old} - \\mu_{new} \\neq 0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a. What is the **conversion rate** for $p_{new}$ under the null? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_new = df2.converted.mean()\n",
    "p_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. What is the **conversion rate** for $p_{old}$ under the null? <br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_old = df2.converted.mean()\n",
    "p_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is $n_{new}$, the number of individuals in the treatment group?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145310"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_new = df2.query(\"group == 'treatment'\").user_id.nunique()\n",
    "n_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is $n_{old}$, the number of individuals in the control group?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145274"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_old = df2.query('group == \"control\"').user_id.nunique()\n",
    "n_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Simulate $n_{new}$ transactions with a conversion rate of $p_{new}$ under the null.  Store these $n_{new}$ 1's and 0's in **new_page_converted**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "    new_page_converted = np.random.choice([0,1], size=n_new, p=[(1-p_new), p_new])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Simulate $n_{old}$ transactions with a conversion rate of $p_{old}$ under the null.  Store these $n_{old}$ 1's and 0's in **old_page_converted**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "    old_page_converted = np.random.choice([0,1], size=n_old, p=[(1-p_old), p_old])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "g. Find $p_{new}$ - $p_{old}$ for your simulated values from part (e) and (f)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0015782389853555567"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_result = df2.query('converted == 1 and landing_page == \"new_page\"')['user_id'].nunique() / n_new\n",
    "old_result = df2.query('converted == 1 and landing_page == \"old_page\"')['user_id'].nunique() / n_old\n",
    "\n",
    "obs_diff = new_result - old_result\n",
    "obs_diff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "h. Create 10,000 $p_{new}$ - $p_{old}$ values using the same simulation process you used in parts (a) through (g) above. Store all 10,000 values in a NumPy array called **p_diffs**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(145310, 145274, 0.11959708724499628, 0.11959708724499628)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_diffs = []\n",
    "size = df2.shape[0]\n",
    "n_new = df2.query('landing_page == \"new_page\"').shape[0]\n",
    "n_old = df2.query('landing_page == \"old_page\"').shape[0]\n",
    "p_new = df2.converted.mean()\n",
    "p_old = df2.converted.mean() \n",
    "n_new, n_old, p_new, p_old"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [],
   "source": [
    "for _ in range(10000):\n",
    "    sample_set = df2.sample(size, replace=True)\n",
    "    new_page_converted = np.random.choice([0,1], size=n_new, p=[(1-p_new), p_new])\n",
    "    old_page_converted = np.random.choice([0,1], size=n_old, p=[(1-p_old), p_old])\n",
    "    p_diffs.append(new_page_converted.mean() - old_page_converted.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "i. Plot a histogram of the **p_diffs**.  Does this plot look like what you expected?  Use the matching problem in the classroom to assure you fully understand what was computed here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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qmq6q6ampqQXtnCTpDUcaAC+1Szu06d5WnwHWDLVbDew5TF2SNCFHGgDbgQNP8mwC7hqqX9GeBjoXeKVdIroXWJ9kZbv5u77VJEkTMu/3AJL8LnAecFKSGQZP81wP3JHkKuB54PLW/B7gYmA38CpwJUBV7U9yHfBwa3dtVR16Y1mStITmDYCq+ugcq86fpW0BV8+xn63A1rF6J0laNP4UhCR1ygCQpE4ZAJLUKQNAkjplAEhSpwwASeqUASBJnTIAJKlTBoAkdcoAkKROGQCS1CkDQJI6ZQBIUqcMAEnq1Lw/By1Jk7Z2y90TOe5z118ykeMuFc8AJKlTBoAkdcoAkKROGQCS1CkDQJI65VNAy8iknpSQdHRa8jOAJBcmeSbJ7iRblvr4kqSBJQ2AJMcA/wW4CDgT+GiSM5eyD5KkgaU+Azgb2F1Vz1bVXwG3ARuWuA+SJJb+HsAq4IWh5RngnCXuw6LzWry0PEzy3/JSfAt5qQMgs9TqoAbJZmBzW/yLJM8scB9OAv50gff5drKcxzfy2H7xwMwNH1m0ziyC5fzegeMbS254S5v/zVEaLXUAzABrhpZXA3uGG1TVzcDNi9WBJDuranqx9j9py3l8y3ls4PiOdkfj+Jb6HsDDwLokpyc5FtgIbF/iPkiSWOIzgKp6Pcm/Au4FjgG2VtWTS9kHSdLAkn8RrKruAe5Z6uMOWbTLS28Ty3l8y3ls4PiOdkfd+FJV87eSJC07/haQJHVqWQRAkhOS7Eiyq01XztFuU2uzK8mmofrPJ/lW+3mKG5NkaN3H2k9XPJnkM0sxnln6vWjja+v/bZJKctJij2U2izW+JL+Z5DtJnkjylSTHL9WY2vEP+7MnSY5Lcntb/2CStUPrPtHqzyS5YNR9LqWFHl+SNUm+keTp9u/t40s3mjdbjPevrTsmyWNJvrr4o5hHVR31L+AzwJY2vwW4YZY2JwDPtunKNr+yrXuIwaPjAf4HcFGr/xLwNeC4tnzychpfW7eGwU35PwJOWk7jA9YDK9r8DbPtdxHHdAzwXeAM4Fjgm8CZh7T5l8AX2vxG4PY2f2ZrfxxwetvPMaPs8ygf36nAWa3NzwD/dzmNb2i7fw38DvDVSYxt+LUszgAY/JzEtja/DbhsljYXADuqan9VvQzsAC5Mcirwrqr6PzV4d24d2v5fANdX1WsAVbV3MQdxGIs1PoDPAr/OIV/IW2KLMr6q+sOqer1t/wCD750slVF+9mR43HcC57ezlw3AbVX1WlV9D9jd9vd2+imVBR9fVb1YVY8CVNWfA08z+PWASViM948kq4FLgC8uwRjmtVwC4JSqehGgTU+epc1sP0Oxqr1mZqkDvAf4u+307n8m+YUF7/loFmV8SS4F/riqvrkYnR7DYr1/w36VwdnBUpmrv7O2aUH1CnDiYbYdZZ9LZTHG92PtcsoHgQcXsM/jWKzxfY7BB64fLXyXx3fU/P8AknwN+OuzrPrkqLuYpVaHqcPgv89K4FzgF4A7kpzRPmkuqKUeX5KfbPteP+L+35IJvX8Hjv1J4HXgSyMeayHM26/DtJmrPtsHtkmduS3G+AYbJT8N/D7wa1X1Z0fcw7dmwceX5CPA3qp6JMl5b7F/C+KoCYCq+uW51iV5KcmpVfViuyQw26WaGeC8oeXVwP2tvvqQ+p6hbb7c/uA/lORHDH7vY9+RjmMuExjf32JwffKb7Z7pauDRJGdX1Z+8haHMakLvH+1m8UeA8xcjuA9j3p89GWozk2QF8G5g/zzbzrfPpbIo40vyDgZ//L9UVV9enK6PZDHGdylwaZKLgXcC70ry21X1jxdnCCOY9E2IhXgBv8nBNxE/M0ubE4DvMfhEv7LNn9DWPczgU/6Bm4gXt/o/B65t8+9hcFqX5TK+Q7Z/jsndBF6s9+9C4ClgagJjWsHgRvXpvHET8X2HtLmag28i3tHm38fBNxGfZXBTct59HuXjC4N7OJ+bxJgWe3yHbHseb4ObwBM9+AK+WScC9wG72vTAH4Zp4ItD7X6VwQ2Z3cCVQ/Vp4NsM7tb/Z974gtyxwG+3dY8CH15O4zvkGM8xuQBYrPdvN4PQfry9vrDE47qYwZMs3wU+2WrXApe2+XcCv9f6+RBwxtC2n2zbPcPBT229aZ+Tei30+IC/w+ASyhND79mbPqwcreM7ZN/n8TYIAL8JLEmdWi5PAUmSxmQASFKnDABJ6pQBIEmdMgAkqVMGgCR1ygCQpE4ZAJLUqf8PVTQ0F8X83l0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(p_diffs); \n",
    "# plot line for observed statistic\n",
    "plt.axvline(obs_diff, c='red');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "j. What proportion of the **p_diffs** are greater than the actual difference observed in **ab_data.csv**?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [],
   "source": [
    "nulls_values = np.random.normal(0, np.std(p_diffs), len(p_diffs))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00012824,  0.00271144, -0.0015227 , ...,  0.00156634,\n",
       "       -0.00034546,  0.00092421])"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nulls_values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.902\n",
      "fail to reject null hypothesis\n"
     ]
    }
   ],
   "source": [
    "p_value =(nulls_values > obs_diff).mean()\n",
    "print(p_value)\n",
    "\n",
    "if p_value < 0.05:\n",
    "    print(\"reject null hypothesis\")\n",
    "else:\n",
    "    print(\"fail to reject null hypothesis\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "k. Please explain using the vocabulary you've learned in this course what you just computed in part **j.**  What is this value called in scientific studies?  What does this value mean in terms of whether or not there is a difference between the new and old pages?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**ANSWER**:\n",
    "\n",
    "\n",
    "In part `j` we calcluated what is known as the **p value** which is a number between `0` and `1` to determine if your null hypothesis is statistically significant which if it is, this allows us to reject the null hypothesis. When \"your null hypothesis is true, the statistical summary would be greater than or equal to the actual observed results\" (cite 1). \n",
    "\n",
    "In this case the **p value** was `.90` which is greater than the `alpha` of `.05` so we deem this as not statistically significant and we fail to reject the null hypothesis. \n",
    "\n",
    "In conclusion, we should go back to our business logic and see if there are any new ways in which we should try other testing techniques such as lengthening the duration of each test period and or including more varied types of users.\n",
    "\n",
    "> Citations:\n",
    "1. https://en.wikipedia.org/wiki/P-value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "l. We could also use a built-in to achieve similar results.  Though using the built-in might be easier to code, the above portions are a walkthrough of the ideas that are critical to correctly thinking about statistical significance. Fill in the below to calculate the number of conversions for each page, as well as the number of individuals who received each page. Let `n_old` and `n_new` refer the the number of rows associated with the old page and new pages, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pandas.core import datetools\n",
    "import statsmodels.api as sm\n",
    "\n",
    "convert_old = df2.query('landing_page == \"old_page\" and converted == 1').shape[0]\n",
    "convert_new = df2.query('landing_page == \"new_page\" and converted == 1').shape[0]\n",
    "\n",
    "n_old = df2.query('landing_page == \"old_page\"').user_id.count()\n",
    "n_new = df2.query('landing_page == \"new_page\"').user_id.count()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "m. Now use `stats.proportions_ztest` to compute your test statistic and p-value.  [Here](http://knowledgetack.com/python/statsmodels/proportions_ztest/) is a helpful link on using the built in."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9050583127590245 -1.3109241984234394\n",
      "fail to reject null hypothesis\n"
     ]
    }
   ],
   "source": [
    "from statsmodels.stats.proportion import proportions_ztest\n",
    "# count : the number of successes in nobs trials. \n",
    "# nobs : number of trials or observations, with the same length as count.\n",
    "\n",
    "count = [convert_new, convert_old]\n",
    "nobs = [n_new, n_old] \n",
    "stat, pval = sm.stats.proportions_ztest(count, nobs, alternative = 'larger')\n",
    "print(float(pval), stat)\n",
    "\n",
    "if pval<0.05:\n",
    "    print(\"reject null hypothesis\")\n",
    "else:\n",
    "    print(\"fail to reject null hypothesis\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "n. What do the z-score and p-value you computed in the previous question mean for the conversion rates of the old and new pages?  Do they agree with the findings in parts **j.** and **k.**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**ANSWER**:\n",
    "\n",
    "\n",
    "The results from our `proportions_ztest` were a **p-value** of `.905` and **z-score** of `-1.31`. \n",
    "\n",
    "The **z-score** is the number of standard deviations our data points are  away from the mean (cite 1). We calculate this by subtracting our data points from the mean from and then dividing by the our standard deviation. In this exercise I am learning about the different types of hypothesis tests we can use on `population parameters`, such as, `t-tests, z-tests, ANOVA and chi-square` tests, to name a few (cite 2). When we use a `z-test` some requirements are random selection, equal sample sizes, normally distributed data, and independent data points. `Z-tests` are used to test whether a `population parameter (σ²)` is significantly different from our hypothesis value and as a formula looks like this (cite 3): $$z = (x – μ) / σ$$ \n",
    "\n",
    "We set up our null and alternate hypothesis as equal or not equal, where the null proposes conversion rate was the same for both users who had landed on our \"old\" page and for users who landed on our \"new\" page. While the alternative proposes either the \"old\" or the \"new\" would lead to more conversions. Our **p-value** in these cases would be the shaded left and right tail regions of our graph as well. \n",
    "\n",
    "Our **p-value** here indicates we stay with our null hypothesis as final outcome.\n",
    "\n",
    "Our **z-score** derived from a one-sided `z-test`, was a negative which means it was below the mean and we could say our conversion rates were lower than average. Rather, also our mean is greater than our data point or even our data point minus our mean is negative. Our area to the left of this negative **z-score** using a `z-table` is `0.0951` derived from the negative `z-table` at `-1.3` and `.01` (cite 5).\n",
    "\n",
    "In part `j`and `k` our **p-value** was `0.4961` and so we failed to reject null hypothesis. Here in this part `m` our p-value was `.905` and we also fail to reject the null hypothesis. In part `k` I noted to possible other ways in which to re-test using different user groups, duration of testing, as possible solutions to getting higher conversion rates.\n",
    "\n",
    "In conclusion, some questioning strategies we learned in lesson 12, video 27 of `Hypothesis Testing` states, \"Is my sample representative of the population to ensure everyone is acurately represented in my sample?, What is the impact of the sample size on my results?\" (cite 6). The instructor notes that in our world today of big data there has been a shift in considering hypothesis testing because sample sizes play a significant factor in results and we need more than just the `average` of sample sizes. Thus, we introduce new approaches that include machine learning techniques either supervised or unsupervised.\n",
    "\n",
    "\n",
    "> Citations: \n",
    "1. https://www.khanacademy.org/math/ap-statistics/density-curves-normal-distribution-ap/measuring-position/v/z-score-introduction\n",
    "2. https://towardsdatascience.com/hypothesis-testing-in-machine-learning-using-python-a0dc89e169ce\n",
    "3. https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/z-score/\n",
    "4. http://www.z-table.com/t-value-table.html\n",
    "5. https://www.thoughtco.com/calculate-probabilities-standard-normal-distribution-table-3126378\n",
    "6. https://classroom.udacity.com/nanodegrees/nd002/parts/bb0cbeb6-d3f8-4bae-9dc3-4abba3823e54/modules/82c7c576-88f5-43b6-80fc-c53378527a2d/lessons/ad8aca80-f444-4479-8848-5b07bafaf66c/concepts/9359bb35-975b-42b6-a8c0-de576c3267f3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='regression'></a>\n",
    "### Part III - A regression approach\n",
    "\n",
    "`1.` In this final part, you will see that the result you achieved in the A/B test in Part II above can also be achieved by performing regression.<br><br> \n",
    "\n",
    "a. Since each row is either a conversion or no conversion, what type of regression should you be performing in this case?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**ANSWER**:\n",
    "\n",
    "From the two main branches of machine learning as `supervised` or `unsupervised`, and given our dataset with a simple encoding of `1` as converted and `0` as not converted, we would move toward `supervised` learning using `linear` and `logistic` regression would be the types of regression used in this case. In our lesson 15, we learned about `multiple linear` regression as creating a matrix of inputs and a vector of the response to predict. In this case, we can compare more than two quantitative variables and even categorical as well thru encoding with dummy variables. This is best visualized as a scatter plot with a slope, intercept and correlation coeffecient."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. The goal is to use **statsmodels** to fit the regression model you specified in part **a.** to see if there is a significant difference in conversion based on which page a customer receives. However, you first need to create in df2 a column for the intercept, and create a dummy variable column for which page each user received.  Add an **intercept** column, as well as an **ab_page** column, which is 1 when an individual receives the **treatment** and 0 if **control**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import LabelEncoder\n",
    "\n",
    "ab_page = LabelEncoder()\n",
    "df2[\"ab_page\"] = ab_page.fit_transform(df2[\"group\"])\n",
    "df2[['control','treatment']]= pd.get_dummies(df2['group'])\n",
    "df2.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Use **statsmodels** to instantiate your regression model on the two columns you created in part b., then fit the model using the two columns you created in part **b.** to predict whether or not an individual converts. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Ordinary Least Squares Regression Approach**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pandas.core import datetools\n",
    "import statsmodels.api as sm\n",
    "\n",
    "df2['intercept'] = 1\n",
    "\n",
    "lm = sm.OLS(df2['converted'], df2[['intercept', 'ab_page']])\n",
    "results = lm.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Logistic Regression Approach**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df2['intercept'] = 1\n",
    "\n",
    "log_mod = sm.Logit(df2['converted'], df2[['intercept', 'ab_page']])\n",
    "results = log_mod.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.exp(-0.0150), 1/np.exp(-0.0150)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. Provide the summary of your model below, and use it as necessary to answer the following questions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "**SUMMARY**:\n",
    "\n",
    "1. Using the first approach of `ordinary least squares regression` our correlation coefficient or `coef` at intercept is `0.1204` which is within the `weak` range in strength and direction of our linear relationship while the `coef` at `ab_page` is `-0.0016` which is a similar `weak` range but, in both cases, not necessarily an indication of a `weak` relationship. We can better see this relationship in a plot to further speculate.\n",
    "\n",
    "    Our slope `ab_page`, is our `explanatory` variable for, `treatment` and `control` which has a `coef` that tells us about our model fit. Here, the predicted conversion is our intercept value multiplied by our slope value which if at `0` would increase by the `coef` value, `-0.0016` over time.\n",
    "\n",
    "    Our `p_value`'s  are given if our parameter for intercept or slope are equal to `0` and the null hypothesis, (cite 1) as we learned in lesson 14, video 16. The `p_value` at our intercept is `0.0` which is statistically significant, while our `ab_page` slope is greater than our alpha of `.05` which suggests not statistically significant for predicting user conversions. \n",
    "\n",
    "    The `rsquared` value is the square of the correlation coefficient and here our value is `0.0` whereas, what we learned in lesson 14, video 17, that the closer our `rsquared` value is to `1` the better fit our model is. So in this case, the model does not fit well.\n",
    "\n",
    "\n",
    "2. With our second approach of `logistic regression` our `coef` at intercept is `-1.988` which is also within the `weak` range in strength and direction while the `coef` at `ab_page` is `-0.0150` which is a similar `weak` range but, again not necessarily a indicator yet.  \n",
    "\n",
    "    Our slope `ab_page`, same as above, has a predicted conversion as our intercept value multiplied by our slope value which if at `0` would increase by the `coef` value, `-0.0150` over time.\n",
    "\n",
    "    Our `p_value` at our intercept is `0.0` which is statistically significant, while our `ab_page` slope is `0.190` which is greater than our alpha of `.05` suggesting not statistically significant for predicting user conversions. \n",
    "    \n",
    "    With `logistic regression` we also have to dive deeper into the `coef`'s and exponentiate each. In this case, our exponentiated `ab_page` value was `0.985` which is the multiplicative change in the odds (cite 2). For each conversion, the group type was `.985` times as likely holding all else constant. When we get values less than `1`, it's good to get the reciprocal as well which in this case is `1.015` and this changes the direction, to a decrease. So then, our interpretation would be for every conversion, our group type is `1.015` times as likely holding all else constant.\n",
    "\n",
    "\n",
    "> Citations: \n",
    "1. https://classroom.udacity.com/nanodegrees/nd002/parts/bb0cbeb6-d3f8-4bae-9dc3-4abba3823e54/modules/82c7c576-88f5-43b6-80fc-c53378527a2d/lessons/d780a3b0-a08e-4282-858e-6a28e8d524aa/concepts/8faca96b-496a-4249-bdb5-0103c5f1a1fb\n",
    "2. lesson 16, video 7: https://classroom.udacity.com/nanodegrees/nd002/parts/bb0cbeb6-d3f8-4bae-9dc3-4abba3823e54/modules/82c7c576-88f5-43b6-80fc-c53378527a2d/lessons/b7ae43e6-204a-4816-a177-6c75115cae5f/concepts/c5af4b47-e3b7-47fb-8941-a4abfcc6a665"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. What is the p-value associated with **ab_page**? Why does it differ from the value you found in **Part II**?<br><br>  **Hint**: What are the null and alternative hypotheses associated with your regression model, and how do they compare to the null and alternative hypotheses in **Part II**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**ANSWER**:\n",
    "\n",
    "Our **p-value** in **Part II** was `0.4961` which is where we failed to reject null hypothesis and similarly, `0.905` for our `z-test` while using our OLS regression model we had a `0.190` and logistic model had a `0.190` value which is still greater than alpha of `0.05`. \n",
    "\n",
    "The reason why this differs from our **Part II** approach, is because with hypothesis testing we are focusing on the population parameters whereas in our regression models and machine learning techniques we focus on users individually and so the **p-value** will differ. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Now, you are considering other things that might influence whether or not an individual converts.  Discuss why it is a good idea to consider other factors to add into your regression model.  Are there any disadvantages to adding additional terms into your regression model?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**ANSWER**:\n",
    "\n",
    "Yes, we should consider other factors that might influence whether or not an individual converts. We have to think about what our model is being used for, and how the inputs are related to a response. Are we looking for a best prediction or looking more for which features or inputs best predict the response? \n",
    "\n",
    "Also, while `linear regression` will provide a `continuous` outcome, `logistic regression` will predict a `discrete` outcome constrained to `0` and `1`. \n",
    "\n",
    "A disadvantage of using a `logistic` regression model is that it can't work with non-linear data points but, rather a decision tree might work better in that case (cite 1). With `logistic` regression we also have to make sure we've clearly defined all of our variables as it can only predict a categorical outcome (cite 1). \n",
    "\n",
    "> Citations: \n",
    "1. https://machinelearning-blog.com/2018/04/23/logistic-regression-101/\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "g. Now along with testing if the conversion rate changes for different pages, also add an effect based on which country a user lives in. You will need to read in the **countries.csv** dataset and merge together your datasets on the appropriate rows.  [Here](https://pandas.pydata.org/pandas-docs/stable/generated/pandas.DataFrame.join.html) are the docs for joining tables. \n",
    "\n",
    "Does it appear that country had an impact on conversion?  Don't forget to create dummy variables for these country columns - **Hint: You will need two columns for the three dummy variables.** Provide the statistical output as well as a written response to answer this question."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "df3 = pd.read_csv('countries.csv')\n",
    "df4 = df2.set_index('user_id').join(df3.set_index('user_id')).head(25)\n",
    "df4.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df4.country.unique()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df4[['UK', 'US', 'CA',]] = pd.get_dummies(df4['country'])\n",
    "df4.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df4.country.value_counts()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "h. Though you have now looked at the individual factors of country and page on conversion, we would now like to look at an interaction between page and country to see if there significant effects on conversion.  Create the necessary additional columns, and fit the new model.  \n",
    "\n",
    "Provide the summary results, and your conclusions based on the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df4['intercept'] = 1\n",
    "\n",
    "logistic_regress = sm.Logit(df4['converted'], df4[['intercept', 'ab_page']])\n",
    "results = logistic_regress.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.exp(results.params)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "1/np.exp(results.params)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df4['intercept'] = 1\n",
    "\n",
    "logistic_regress = sm.Logit(df4['converted'], df4[['intercept', 'UK', 'US']])\n",
    "results = logistic_regress.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.exp(results.params)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "1/np.exp(results.params)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df4['intercept'] = 1\n",
    "\n",
    "logistic_regress = sm.Logit(df4['converted'], df4[['intercept', 'CA', 'US']])\n",
    "results = logistic_regress.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.exp(results.params)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "1/np.exp(results.params)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**SUMMARY**:\n",
    "\n",
    "With our new model including country encodings, we fit three `logistic` regression models with our independent variables, `ab_page`, `UK`, `CA` and `US`.\n",
    "\n",
    "1. For the first scenario, `ab_page` had a `coef` of `0.3747` which very slightly close to a `weak` relationship with a `p_value` of `.703` suggesting we fail to reject the null hypothesis. Prediction of our conversion rates shows `ab_page` group types were 1.45 times more likely holding all other variables constant.  \n",
    "\n",
    "2. For the second scenario,  `UK`, `US`, had `coef` values of  `2.3`, and  `0.51` which are also close to a `weak` relationship while \n",
    "`p_value`’s were `.095` and `.698` respectively both suggesting to reject the null hypothesis. Prediction of our conversion rates shows users in `UK` we’re `10` more times likely than while `US` users were `1.6` times more likely holding all other variables constant.  \n",
    " \n",
    "3. For the third scenario,  `CA`, `US`, had `coef` values of  `-2.3`, and  `-1.79 ` which are close to a `weak` relationship while  `p_value`’s were `.095` and `.287` respectively both suggesting to reject the null hypothesis. However, `CA` is just on the edge of being statistically significant. Maybe there is more we can look into there. Prediction of our conversion rates shows users in `CA` we’re `10` more times likely than while `US` users were `6` times more likely holding all other variables constant.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from subprocess import call\n",
    "call(['python', '-m', 'nbconvert', 'Analyze_ab_test_results_notebook.ipynb'])"
   ]
  }
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