analyze_ab_test_results_notebook.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analyze A/B Test Results\n",
    "\n",
    "You may either submit your notebook through the workspace here, or you may work from your local machine and submit through the next page.  Either way assure that your code passes the project [RUBRIC](https://review.udacity.com/#!/projects/37e27304-ad47-4eb0-a1ab-8c12f60e43d0/rubric).  **Please save regularly\n",
    "\n",
    "This project will assure you have mastered the subjects covered in the statistics lessons.  The hope is to have this project be as comprehensive of these topics as possible.  Good luck!\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",
    "\n",
    "<a id='intro'></a>\n",
    "### Introduction\n",
    "\n",
    "A/B tests are very commonly performed by data analysts and data scientists.  It is important that you get some practice working with the difficulties of these \n",
    "\n",
    "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",
    "**As you work through this notebook, follow along in the classroom and answer the corresponding quiz questions associated with each question.** The labels for each classroom concept are provided for each question.  This will assure you are on the right track as you work through the project, and you can feel more confident in your final submission meeting the criteria.  As a final check, assure you meet all the criteria on the [RUBRIC](https://review.udacity.com/#!/projects/37e27304-ad47-4eb0-a1ab-8c12f60e43d0/rubric).\n",
    "\n",
    "<a id='probability'></a>\n",
    "#### Part I - Probability\n",
    "\n",
    "To get started, let's import our libraries."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "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",
    "#We are setting the seed to assure you get the same answers on quizzes as we set up\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": 3,
   "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>0</th>\n",
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       "      <td>control</td>\n",
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       "      <td>1</td>\n",
       "    </tr>\n",
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       "</table>\n",
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      ],
      "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": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv('ab_data.csv')\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Use the below cell to find the number of rows in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(294478, 5)"
      ]
     },
     "execution_count": 4,
     "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": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290584"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.user_id.nunique()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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       "      <th>2</th>\n",
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       "      <td>control</td>\n",
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      "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": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "d. The proportion of users converted."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11965919355605512"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df['converted'].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "e. The number of times the `new_page` and `treatment` don't line up."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "d = df.query('group == \"treatment\" or landing_page == \"new_page\"')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
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       "      <td>2017-01-11 15:11:16.407599</td>\n",
       "      <td>control</td>\n",
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       "      <th>308</th>\n",
       "      <td>857184</td>\n",
       "      <td>2017-01-20 07:34:59.832626</td>\n",
       "      <td>treatment</td>\n",
       "      <td>old_page</td>\n",
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       "      <th>327</th>\n",
       "      <td>686623</td>\n",
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       "      <td>treatment</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>357</th>\n",
       "      <td>856078</td>\n",
       "      <td>2017-01-12 12:29:30.354835</td>\n",
       "      <td>treatment</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
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      ],
      "text/plain": [
       "     user_id                   timestamp      group landing_page  converted\n",
       "22    767017  2017-01-12 22:58:14.991443    control     new_page          0\n",
       "240   733976  2017-01-11 15:11:16.407599    control     new_page          0\n",
       "308   857184  2017-01-20 07:34:59.832626  treatment     old_page          0\n",
       "327   686623  2017-01-09 14:26:40.734775  treatment     old_page          0\n",
       "357   856078  2017-01-12 12:29:30.354835  treatment     old_page          0"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = d.query('group != \"treatment\" or landing_page != \"new_page\"')\n",
    "d.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Do any of the rows have missing values?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "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": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.isnull().sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`2.` For the rows where **treatment** is not aligned with **new_page** or **control** is not aligned with **old_page**, we cannot be sure if this row truly received the new or old page.  Use **Quiz 2** in the classroom to provide 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": 11,
   "metadata": {},
   "outputs": [
    {
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       "      <th>6</th>\n",
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       "      <td>2017-01-19 03:26:46.940749</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
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      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted\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",
       "6   679687  2017-01-19 03:26:46.940749  treatment     new_page          1\n",
       "8   817355  2017-01-04 17:58:08.979471  treatment     new_page          1\n",
       "9   839785  2017-01-15 18:11:06.610965  treatment     new_page          1"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_treatment = df.query('group == \"treatment\" and landing_page == \"new_page\"') \n",
    "df_control = df.query('group == \"control\" and landing_page == \"old_page\"')\n",
    "\n",
    "df2 = pd.concat([df_treatment, df_control])\n",
    "df2.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 12,
     "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": "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": {
      "text/plain": [
       "2893    773192\n",
       "Name: user_id, dtype: int64"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['user_id'][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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       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\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>1899</th>\n",
       "      <td>773192</td>\n",
       "      <td>2017-01-09 05:37:58.781806</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <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",
       "</div>"
      ],
      "text/plain": [
       "      user_id                   timestamp      group landing_page  converted\n",
       "1899   773192  2017-01-09 05:37:58.781806  treatment     new_page          0\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(keep=False)]"
   ]
  },
  {
   "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": [
    {
     "data": {
      "text/plain": [
       "(290584, 5)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2 = df2.drop_duplicates(subset='user_id')\n",
    "df2.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`4.` Use **df2** in the below cells 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": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.converted.mean()"
   ]
  },
  {
   "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": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1203863045004612"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_control_converted = df2.query('group == \"control\"')['converted'].mean()\n",
    "p_control_converted"
   ]
  },
  {
   "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": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11880806551510564"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_treatment_converted = df2.query('group == \"treatment\"')['converted'].mean()\n",
    "p_treatment_converted"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is the probability that an individual received the new page?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5000619442226688"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(df2.query('landing_page == \"new_page\"')) / len(df2['landing_page'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Use the results in the previous two portions of this question to suggest if you think there is evidence that one page leads to more conversions?  Write your response below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:** There is some evidence to suggest that one that the old page leads to more conversions than the new page when comparing the conversion mean of the treatment and control groups. "
   ]
  },
  {
   "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": [
    "**Null Hypothesis:** We assume that the new treatment has no effect on the probability of conversion.\n",
    "\n",
    "**Alternative Hypothesis:** The new treatment will increase the probability of conversion.\n",
    "\n",
    "$H_{0}$: $p_{old}$ >= $p_{new}$\n",
    "\n",
    "$H_{1}$: $p_{old}$ < $p_{new}$"
   ]
  },
  {
   "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": [
    "a. What is the **convert rate** for $p_{new}$ under the null? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_new = df2['converted'].mean()\n",
    "p_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. What is the **convert rate** for $p_{old}$ under the null? <br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_old = df2['converted'].mean()\n",
    "p_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is $n_{new}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145310"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_new = df2.query('landing_page == \"new_page\"')['landing_page'].count()\n",
    "n_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is $n_{old}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145274"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_old = df2.query('landing_page == \"old_page\"')['landing_page'].count()\n",
    "n_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Simulate $n_{new}$ transactions with a convert 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": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11911086642350836"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_page_converted = np.random.choice([1, 0], size=n_new, p=[p_new, (1-p_new)])\n",
    "new_page_converted.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Simulate $n_{old}$ transactions with a convert 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": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.12128804879056128"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "old_page_converted = np.random.choice([1, 0], size=n_old, p=[p_old, (1-p_old)])\n",
    "old_page_converted.mean()"
   ]
  },
  {
   "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": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_new - p_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "h. Simulate 10,000 $p_{new}$ - $p_{old}$ values using this same process similarly to the one you calculated in parts **a. through g.** above.  Store all 10,000 values in **p_diffs**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "p_diffs = [] #Simulation 10,000 to calculate the differences in probability between the old and new page\n",
    "\n",
    "for _ in range(10000):\n",
    "    \n",
    "    new_page_converted = np.random.choice([1, 0], size=n_new, p=[p_new, (1-p_new)]).mean()\n",
    "    old_page_converted = np.random.choice([1, 0], size=n_old, p=[p_old, (1-p_old)]).mean()\n",
    "    difference = new_page_converted - old_page_converted \n",
    "    p_diffs.append(difference)"
   ]
  },
  {
   "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": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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dDJwBXCnpk8CfgCPzstnAFGARsAo4DiAilkv6KjA31/tKRCyvMG4zM+tFZYkjIm6j8fgEwAEN6gdwQg/bmgnMHLjozMysv/zNcTMzK8WJw8zMSnHiMDOzUpw4zMysFCcOMzMrxYnDzMxKceIwM7NSnDjMzKwUJw4zMyulqcQhafeqAzEzs87Q7BnHdyTdKel4SVtXGpGZmbW1phJHROwHHEO6vfk8SZdLOqjSyMzMrC01PcYREQuBLwMnAe8GzpP0oKS/rSo4MzNrP82OcbxZ0tmk3w3/G+B9EfGmPH12hfGZmVmbafa26t8CLgROjogXa4URsVTSlyuJzMzM2lKziWMK8GJEvAIgaQNgk4hYFRGXVhadmZm1nWbHOG4GNi3Mb5bLzMxsiGk2cWwSEc/XZvL0ZtWEZGZm7azZxPGCpL1qM5LeCrzYS30zM1tPNTvG8VngKklL8/wY4MPVhGRDyYQZN7Q6BDMrqanEERFzJe0K7AIIeDAi/lJpZGZm1paaPeMAeBswIa+zpyQi4pJKojIzs7bVVOKQdCmwM3AP8EouDsCJw8xsiGn2jGMyMCkiospgzMys/TV7VdV9wBuqDMTMzDpDs2cco4D7Jd0JvFQrjIgjKonKzMzaVrOJ47QqgzAzs87R7OW4v5a0AzAxIm6WtBkwrNrQzMysHTV7W/VPAz8BvpuLxgI/7WOdmZKelHRfoew0SY9Juic/phSWfVHSIkkPSTqkUH5oLlskaUaZgzMzs4HXbFfVCcDewB2QftRJ0jZ9rHMR6Xbs9Zfsnh0R3ygWSJoEHAXsBmwH3CzpjXnxt4GDgC5grqRZEXF/k3GbWdbKb+kvPuPwlu3bBl6zV1W9FBEv12YkDSd9j6NHEXErsLzJ7U8FroiIlyLiEWARKVHtDSyKiIfz/q/Idc3MrEWaTRy/lnQysGn+rfGrgJ/1c58nSpqfu7JG5LKxwJJCna5c1lP5WiRNlzRP0rzu7u5+hmZmZn1pNnHMALqBe4G/A2aTfn+8rPNJ30DfA1gGfDOXq0Hd6KV87cKICyJickRMHj16dD9CMzOzZjR7VdWrpJ+OvXBddhYRT9SmJV0IXJ9nu4DxharjgNqdeHsqNzOzFmj2XlWP0OCTfkTsVGZnksZExLI8+wHSN9IBZgGXSzqLNDg+EbiTdMYxUdKOwGOkAfSPlNmnmZkNrDL3qqrZBDgSGNnbCpJ+BOwPjJLUBZwK7C9pD1ISWkzq9iIiFki6ErgfWA2cUPh98xOBm0jfG5kZEQuajNnMzCrQbFfV03VF50i6DTill3WOblD8/V7qnw6c3qB8NmlMxczM2kCzXVV7FWY3IJ2BbFlJRGZm1taa7ar6ZmF6Namb6UMDHo2ZmbW9Zruq3lN1IGZm1hma7ar6p96WR8RZAxOOmZm1uzJXVb2NdNkswPuAW1nzW91mZjYElPkhp70iYiWku9wCV0XEp6oKzMzM2lOztxzZHni5MP8yMGHAozEzs7bX7BnHpcCdkq4lfXnvA6x9u3QzMxsCmr2q6nRJPwfelYuOi4jfVReWmZm1q2a7qgA2A1ZExLlAV75/lJmZDTHN/nTsqcBJwBdz0YbAZVUFZWZm7avZM44PAEcALwBExFJ8yxEzsyGp2cTxckQE+dbqkjavLiQzM2tnzSaOKyV9F9ha0qeBm1nHH3UyM7PO1OxVVd/IvzW+AtgFOCUi5lQamZmZtaU+E4ekYcBNEXEg4GRhZjbE9dlVlX+Jb5Wk1w1CPGZm1uaa/eb4n4F7Jc0hX1kFEBH/WElUZmbWtppNHDfkh5mZDXG9Jg5J20fEnyLi4sEKyMzM2ltfYxw/rU1IurriWMzMrAP0lThUmN6pykDMzKwz9JU4oodpMzMbovoaHH+LpBWkM49N8zR5PiJiq0qjMzOzttNr4oiIYYMViJmZdYYyv8dhZmbmxGFmZuU4cZiZWSmVJQ5JMyU9Kem+QtlISXMkLczPI3K5JJ0naZGk+ZL2KqwzLddfKGlaVfGamVlzqjzjuAg4tK5sBnBLREwEbsnzAIcBE/NjOnA+pEQDnArsA+wNnFpLNmZm1hqVJY6IuBVYXlc8FajdvuRi4P2F8ksiuZ30g1FjgEOAORGxPCKeId3WvT4ZmZnZIBrsMY5tI2IZQH7eJpePBZYU6nXlsp7K1yJpuqR5kuZ1d3cPeOBmZpa0y+C4GpRFL+VrF0ZcEBGTI2Ly6NGjBzQ4MzN7zWAnjidyFxT5+clc3gWML9QbByztpdzMzFpksBPHLKB2ZdQ04LpC+cfz1VX7As/lrqybgIMljciD4gfnMjMza5Fmf8ipNEk/AvYHRknqIl0ddQZwpaRPAn8CjszVZwNTgEXAKuA4gIhYLumrwNxc7ysRUT/gbmZmg6iyxBERR/ew6IAGdQM4oYftzARmDmBoZma2DtplcNzMzDqEE4eZmZXixGFmZqU4cZiZWSlOHGZmVooTh5mZleLEYWZmpThxmJlZKU4cZmZWihOHmZmV4sRhZmalOHGYmVkpThxmZlaKE4eZmZXixGFmZqU4cZiZWSlOHGZmVooTh5mZleLEYWZmpThxmJlZKU4cZmZWihOHmZmV4sRhZmalOHGYmVkpThxmZlaKE4eZmZUyvNUBWHuYMOOGVodgZh3CZxxmZlZKSxKHpMWS7pV0j6R5uWykpDmSFubnEblcks6TtEjSfEl7tSJmMzNLWtlV9Z6IeKowPwO4JSLOkDQjz58EHAZMzI99gPPzs5l1iFZ1hS4+4/CW7Hd9105dVVOBi/P0xcD7C+WXRHI7sLWkMa0I0MzMWpc4AviFpLskTc9l20bEMoD8vE0uHwssKazblcvWIGm6pHmS5nV3d1cYupnZ0Naqrqp3RsRSSdsAcyQ92EtdNSiLtQoiLgAuAJg8efJay83MbGC05IwjIpbm5yeBa4G9gSdqXVD5+clcvQsYX1h9HLB08KI1M7OiQU8ckjaXtGVtGjgYuA+YBUzL1aYB1+XpWcDH89VV+wLP1bq0zMxs8LWiq2pb4FpJtf1fHhE3SpoLXCnpk8CfgCNz/dnAFGARsAo4bvBDNjOzmkFPHBHxMPCWBuVPAwc0KA/ghEEIzczMmtBOl+OamVkHcOIwM7NSnDjMzKwUJw4zMyvFicPMzEpx4jAzs1KcOMzMrBQnDjMzK8WJw8zMSnHiMDOzUpw4zMysFCcOMzMrxYnDzMxKceIwM7NSnDjMzKwUJw4zMyvFicPMzEpx4jAzs1KcOMzMrBQnDjMzK8WJw8zMSnHiMDOzUoa3OgB7zYQZN7Q6BDOzPvmMw8zMSnHiMDOzUtxVZWbrrVZ2/y4+4/CW7btqPuMwM7NSnDjMzKyUjkkckg6V9JCkRZJmtDoeM7OhqiMSh6RhwLeBw4BJwNGSJrU2KjOzoalTBsf3BhZFxMMAkq4ApgL3V7Ezf5/CzNZVq95HBmNQvlMSx1hgSWG+C9inWEHSdGB6nn1e0kODFFvRKOCpFuy3Xbk91uY2WZPbY03r3B46c532v0MzlTolcahBWawxE3EBcMHghNOYpHkRMbmVMbQTt8fa3CZrcnusqVPaoyPGOEhnGOML8+OApS2KxcxsSOuUxDEXmChpR0kbAUcBs1ock5nZkNQRXVURsVrSicBNwDBgZkQsaHFYjbS0q6wNuT3W5jZZk9tjTR3RHoqIvmuZmZllndJVZWZmbcKJw8zMSnHiaIKkkZLmSFqYn0f0UG9arrNQ0rRC+Vsl3Ztvl3KeJNWt93lJIWlU1ccyEKpqD0n/IelBSfMlXStp68E6pv7o6zY4kjaW9OO8/A5JEwrLvpjLH5J0SLPbbGcD3R6Sxkv6paQHJC2Q9JnBO5qBUcXfSF42TNLvJF1f/VE0EBF+9PEA/h2YkadnAGc2qDMSeDg/j8jTI/KyO4G3k76P8nPgsMJ640mD/o8Co1p9rK1sD+BgYHiePrPRdtvlQbpI44/ATsBGwO+BSXV1jge+k6ePAn6cpyfl+hsDO+btDGtmm+36qKg9xgB75TpbAn/olPaoqk0K6/0TcDlwfSuOzWcczZkKXJynLwbe36DOIcCciFgeEc8Ac4BDJY0BtoqI30Z6xS+pW/9s4AvUfaGxzVXSHhHxi4hYnde/nfR9nXb1P7fBiYiXgdptcIqK7fQT4IB8djUVuCIiXoqIR4BFeXvNbLNdDXh7RMSyiLgbICJWAg+Q7iLRKar4G0HSOOBw4HuDcAwNOXE0Z9uIWAaQn7dpUKfRbVHG5kdXg3IkHQE8FhG/ryLoClXSHnU+QTobaVc9HV/DOjkhPge8vpd1m9lmu6qiPf5H7sLZE7hjAGOuWlVtcg7pw+arAx9yczriexyDQdLNwBsaLPpSs5toUBY9lUvaLG/74Ca3P6gGuz3q9v0lYDXwwyb31Qp9HkcvdXoqb/RBrlPORKtoj7SStAVwNfDZiFjR7wgH34C3iaT3Ak9GxF2S9l/H+PrNiSOLiAN7WibpCUljImJZ7mp5skG1LmD/wvw44Fe5fFxd+VJgZ1Lf5e/z2PA44G5Je0fE4+twKAOiBe1R2/Y04L3AAbkrq101cxucWp0uScOB1wHL+1i3U2+tU0l7SNqQlDR+GBHXVBN6ZapokyOAIyRNATYBtpJ0WUR8tJpD6EGrB5A64QH8B2sOBv97gzojgUdIA8Ej8vTIvGwusC+vDQZPabD+YjpncLyS9gAOJd0qf3Srj7GJNhhOGvDfkdcGPnerq3MCaw58Xpmnd2PNgc+HSQOpfW6zXR8VtYdIY2DntPr42qVN6tbdnxYNjre8cTvhQepzvAVYmJ9rb4CTge8V6n2CNIi1CDiuUD4ZuI90ZcS3yN/Yr9tHJyWOStoj11sC3JMf32n1sfbRDlNIV/r8EfhSLvsKcESe3gS4Kh/XncBOhXW/lNd7iDWvsltrm53yGOj2APYjddvML/xNrPWhq50fVfyNFJa3LHH4liNmZlaKr6oyM7NSnDjMzKwUJw4zMyvFicPMzEpx4jAzs1KcOMzMrBQnDrNBImn/2m2wJR1Ru822pNH5ltq/k/QuSUfmW4n/srURmzXmW46YtUBEzAJm5dkDgAcjYhqApBuB4yPCicPakhOHWUn5Tq03ku7Uuifpm8Efj4hVDeoeSrqb6VPA3YXyY8nftCf9vsmmku4BriV9Y3pHSbOAi4AfkG5ZsQHwwYhYWM2RmTXHXVVm/bMLcEFEvBlYQfpBnjVI2gS4EHgf8C4a3G04Iu4BTiH9gM8eEfGvwDzgmIj4Z+DvgXMjYg9Soumq34bZYHPiMOufJRHxmzx9Geksod6uwCMRsTDSvX0u68d+fgucLOkkYIeIeLF/4ZoNHCcOs/6pv8lbTzd9W6ebwUXE5aRbab8I3CTpb9Zle2YDwYnDrH+2l/T2PH00cFuDOg+Sxip2LtQrRdJOwMMRcR5pMP3N/QnWbCA5cZj1zwPANEnzSb89cn59hYj4MzAduEHSbcCj/djPh4H78sD5rqTfpzBrKd9W3aykfFXV9RGxe4tDMWsJn3GYmVkpPuMwGwCSriX9xGfRSRFxUyviMauSE4eZmZXiriozMyvFicPMzEpx4jAzs1KcOMzMrJT/Bv3YI66O9crFAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe54c6f0cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(p_diffs)\n",
    "plt.xlabel('p_diffs')\n",
    "plt.ylabel('Frequency')\n",
    "plt.title('10k Simulated Probability Differences');"
   ]
  },
  {
   "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": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0014795997940775518"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#calculation of actual differences\n",
    "act_diff = df[df['group'] == 'treatment']['converted'].mean() -  df[df['group'] == 'control']['converted'].mean()\n",
    "act_diff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  8.03796729e-05,  -5.04549342e-04,   1.00966496e-03, ...,\n",
       "        -8.48690739e-04,   1.92510471e-03,   3.48769682e-04])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#differences based on simulation\n",
    "p_diffs = np.array(p_diffs)\n",
    "p_diffs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.89470000000000005"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#verification - mean of p_diffs that are larger than the actual difference\n",
    "(act_diff < p_diffs).mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "k. In words, explain 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",
    "In the previous exercises, we computed the p-values. A p-value is the probability of observing our statistic if the null hypothesis is true. A larger p-value typically indicates that the null hypothesis is true. If the p-value is smaller than the significance level or the suggested p-critical value, we typically classify the result as significant and reject the null hypothesis.\n",
    "\n",
    "In this situation, our p-value is larger than the p-critical value, therefore we fail to reject the null hypothesis and conclude that the new pages have no impact on the probability of conversion and that we should therefore remain using the old pages."
   ]
  },
  {
   "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": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/conda/lib/python3.6/site-packages/statsmodels/compat/pandas.py:56: FutureWarning: The pandas.core.datetools module is deprecated and will be removed in a future version. Please use the pandas.tseries module instead.\n",
      "  from pandas.core import datetools\n"
     ]
    }
   ],
   "source": [
    "import statsmodels.api as sm\n",
    "\n",
    "convert_old = df2.query('landing_page == \"old_page\"')['converted'].sum()\n",
    "convert_new = df2.query('landing_page == \"new_page\"')['converted'].sum()\n",
    "n_old = len(df2.query('landing_page == \"old_page\"'))\n",
    "n_new = len(df2.query('landing_page == \"new_page\"'))"
   ]
  },
  {
   "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": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.31092419842 0.905058312759\n"
     ]
    }
   ],
   "source": [
    "z_score, p_value = sm.stats.proportions_ztest([convert_old, convert_new], [n_old, n_new], alternative='smaller')\n",
    "print(z_score, p_value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.905058312759\n",
      "1.64485362695\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "\n",
    "print(norm.cdf(z_score))\n",
    "#z-score significance\n",
    "\n",
    "print(norm.ppf(1-(0.05)))\n",
    "#critical value at 95% confidance"
   ]
  },
  {
   "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",
    "The z-score of 1.31092419842 is less than the critical value of 1.64485362695, therefore, we accept the null hypothesis. These scores agree with our previous findings. \n"
   ]
  },
  {
   "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 acheived in the previous A/B test can also be acheived 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:** Logistic Regression"
   ]
  },
  {
   "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 a colun 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": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
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       "\n",
       "    .dataframe thead th {\n",
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       "    }\n",
       "\n",
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       "    }\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",
       "      <th>intercept</th>\n",
       "      <th>treatment</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\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",
       "      <td>2017-01-08 18:28:03.143765</td>\n",
       "      <td>treatment</td>\n",
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       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>679687</td>\n",
       "      <td>2017-01-19 03:26:46.940749</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>817355</td>\n",
       "      <td>2017-01-04 17:58:08.979471</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>839785</td>\n",
       "      <td>2017-01-15 18:11:06.610965</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted  \\\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",
       "6   679687  2017-01-19 03:26:46.940749  treatment     new_page          1   \n",
       "8   817355  2017-01-04 17:58:08.979471  treatment     new_page          1   \n",
       "9   839785  2017-01-15 18:11:06.610965  treatment     new_page          1   \n",
       "\n",
       "   intercept  treatment  \n",
       "2          1          1  \n",
       "3          1          1  \n",
       "6          1          1  \n",
       "8          1          1  \n",
       "9          1          1  "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['intercept']= 1\n",
    "df2[['control', 'treatment']] = pd.get_dummies(df['group'])\n",
    "df2 = df2.drop(['control'], axis=1)\n",
    "df2.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Use **statsmodels** to import your regression model.  Instantiate the model, and fit the model using the two columns you created in part **b.** to predict whether or not an individual converts."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statsmodels.api as sm\n",
    "logit = sm.Logit(df2['converted'],df2[['intercept','treatment']])"
   ]
  },
  {
   "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": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366118\n",
      "         Iterations 6\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>     <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>               <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290582</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>               <td>MLE</td>       <th>  Df Model:          </th>   <td>     1</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Tue, 17 Apr 2018</td> <th>  Pseudo R-squ.:     </th>  <td>8.077e-06</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>02:12:29</td>     <th>  Log-Likelihood:    </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>converged:</th>           <td>True</td>       <th>  LL-Null:           </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th> </th>                      <td> </td>        <th>  LLR p-value:       </th>   <td>0.1899</td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>intercept</th> <td>   -1.9888</td> <td>    0.008</td> <td> -246.669</td> <td> 0.000</td> <td>   -2.005</td> <td>   -1.973</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>treatment</th> <td>   -0.0150</td> <td>    0.011</td> <td>   -1.311</td> <td> 0.190</td> <td>   -0.037</td> <td>    0.007</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                           Logit Regression Results                           \n",
       "==============================================================================\n",
       "Dep. Variable:              converted   No. Observations:               290584\n",
       "Model:                          Logit   Df Residuals:                   290582\n",
       "Method:                           MLE   Df Model:                            1\n",
       "Date:                Tue, 17 Apr 2018   Pseudo R-squ.:               8.077e-06\n",
       "Time:                        02:12:29   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "                                        LLR p-value:                    0.1899\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -1.9888      0.008   -246.669      0.000      -2.005      -1.973\n",
       "treatment     -0.0150      0.011     -1.311      0.190      -0.037       0.007\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results = logit.fit()\n",
    "results.summary()"
   ]
  },
  {
   "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 the **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 the **Part II**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:**\n",
    "\n",
    "$H_{0}$: $p_{new}$ - $p_{old}$ = 0\n",
    "\n",
    "$H_{1}$: $p_{new}$ -$p_{old}$ != 0\n",
    "\n",
    "We are currently doing a two-sided test, however, in Par 2 the test we performed was one-sided. In this test, we must test as to whether $p_{new}$ and $p_{old}$ = 0 are equal or not. "
   ]
  },
  {
   "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",
    "- We should consider other factors into the regression models, as there are a multitude of other factors that could also influence conversions. Some examples that come to mind are whether this a new or returning customer, time of year, and age;\n",
    "- We also need to be careful and account for change aversion and novelty, as this could influence the scores;\n",
    "- The problem with adding additional items to a regression model is that humans are complex beings, and what influences people on an individual level can be very different from one person to another, and is very subjective. It would be impossible to consider every single factor that could possibly influence someone, and at a certain point, we risk overfitting the model. The rate of return also goes down after a certain point. Plus, if you wait for the model to be perfect before acting, this in and of itself is also a decision and you risk missing out on opportunities; "
   ]
  },
  {
   "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. You will need to read in the **countries.csv** dataset and merge together your datasets on the approporiate 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 varaibles.** Provide the statistical output as well as a written response to answer this question."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "      <th></th>\n",
       "      <th>country</th>\n",
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       "      <th>834778</th>\n",
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       "      <td>UK</td>\n",
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       "      <td>UK</td>\n",
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       "      <td>UK</td>\n",
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      ],
      "text/plain": [
       "        country\n",
       "user_id        \n",
       "834778       UK\n",
       "928468       US\n",
       "822059       UK\n",
       "711597       UK\n",
       "710616       UK"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_countries = pd.read_csv('./countries.csv')\n",
    "df_countries = df_countries.set_index('user_id')\n",
    "df_countries.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
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       "      <th></th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "      <th>intercept</th>\n",
       "      <th>treatment</th>\n",
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       "    <tr>\n",
       "      <th>user_id</th>\n",
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       "      <th>661590</th>\n",
       "      <td>2017-01-11 16:55:06.154213</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>853541</th>\n",
       "      <td>2017-01-08 18:28:03.143765</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>679687</th>\n",
       "      <td>2017-01-19 03:26:46.940749</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>817355</th>\n",
       "      <td>2017-01-04 17:58:08.979471</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>839785</th>\n",
       "      <td>2017-01-15 18:11:06.610965</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                          timestamp      group landing_page  converted  \\\n",
       "user_id                                                                  \n",
       "661590   2017-01-11 16:55:06.154213  treatment     new_page          0   \n",
       "853541   2017-01-08 18:28:03.143765  treatment     new_page          0   \n",
       "679687   2017-01-19 03:26:46.940749  treatment     new_page          1   \n",
       "817355   2017-01-04 17:58:08.979471  treatment     new_page          1   \n",
       "839785   2017-01-15 18:11:06.610965  treatment     new_page          1   \n",
       "\n",
       "         intercept  treatment  \n",
       "user_id                        \n",
       "661590           1          1  \n",
       "853541           1          1  \n",
       "679687           1          1  \n",
       "817355           1          1  \n",
       "839785           1          1  "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2 = df2.set_index('user_id')\n",
    "df2.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <td>1</td>\n",
       "      <td>CA</td>\n",
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       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>UK</td>\n",
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       "      <th>839785</th>\n",
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       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>CA</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                          timestamp      group landing_page  converted  \\\n",
       "user_id                                                                  \n",
       "661590   2017-01-11 16:55:06.154213  treatment     new_page          0   \n",
       "853541   2017-01-08 18:28:03.143765  treatment     new_page          0   \n",
       "679687   2017-01-19 03:26:46.940749  treatment     new_page          1   \n",
       "817355   2017-01-04 17:58:08.979471  treatment     new_page          1   \n",
       "839785   2017-01-15 18:11:06.610965  treatment     new_page          1   \n",
       "\n",
       "         intercept  treatment country  \n",
       "user_id                                \n",
       "661590           1          1      US  \n",
       "853541           1          1      US  \n",
       "679687           1          1      CA  \n",
       "817355           1          1      UK  \n",
       "839785           1          1      CA  "
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_merged = pd.merge(df2, df_countries, how='inner', left_index=True, right_index=True)\n",
    "df_merged.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "US    203619\n",
       "UK     72466\n",
       "CA     14499\n",
       "Name: country, dtype: int64"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_merged['country'].value_counts()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <td>1</td>\n",
       "      <td>UK</td>\n",
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       "      <td>1</td>\n",
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       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                          timestamp      group landing_page  converted  \\\n",
       "user_id                                                                  \n",
       "661590   2017-01-11 16:55:06.154213  treatment     new_page          0   \n",
       "853541   2017-01-08 18:28:03.143765  treatment     new_page          0   \n",
       "679687   2017-01-19 03:26:46.940749  treatment     new_page          1   \n",
       "817355   2017-01-04 17:58:08.979471  treatment     new_page          1   \n",
       "839785   2017-01-15 18:11:06.610965  treatment     new_page          1   \n",
       "\n",
       "         intercept  treatment country  US  UK  \n",
       "user_id                                        \n",
       "661590           1          1      US   1   0  \n",
       "853541           1          1      US   1   0  \n",
       "679687           1          1      CA   0   0  \n",
       "817355           1          1      UK   0   1  \n",
       "839785           1          1      CA   0   0  "
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_merged[['US', 'UK']] = pd.get_dummies(df_merged['country'])[['US','UK']]\n",
    "df_merged.head()"
   ]
  },
  {
   "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": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366116\n",
      "         Iterations 6\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>     <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>               <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290581</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>               <td>MLE</td>       <th>  Df Model:          </th>   <td>     2</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Tue, 17 Apr 2018</td> <th>  Pseudo R-squ.:     </th>  <td>1.521e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>02:15:01</td>     <th>  Log-Likelihood:    </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>converged:</th>           <td>True</td>       <th>  LL-Null:           </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th> </th>                      <td> </td>        <th>  LLR p-value:       </th>   <td>0.1984</td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "          <td></td>             <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>intercept_country</th> <td>   -2.0375</td> <td>    0.026</td> <td>  -78.364</td> <td> 0.000</td> <td>   -2.088</td> <td>   -1.987</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>US</th>                <td>    0.0408</td> <td>    0.027</td> <td>    1.518</td> <td> 0.129</td> <td>   -0.012</td> <td>    0.093</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>UK</th>                <td>    0.0507</td> <td>    0.028</td> <td>    1.786</td> <td> 0.074</td> <td>   -0.005</td> <td>    0.106</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                           Logit Regression Results                           \n",
       "==============================================================================\n",
       "Dep. Variable:              converted   No. Observations:               290584\n",
       "Model:                          Logit   Df Residuals:                   290581\n",
       "Method:                           MLE   Df Model:                            2\n",
       "Date:                Tue, 17 Apr 2018   Pseudo R-squ.:               1.521e-05\n",
       "Time:                        02:15:01   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "                                        LLR p-value:                    0.1984\n",
       "=====================================================================================\n",
       "                        coef    std err          z      P>|z|      [0.025      0.975]\n",
       "-------------------------------------------------------------------------------------\n",
       "intercept_country    -2.0375      0.026    -78.364      0.000      -2.088      -1.987\n",
       "US                    0.0408      0.027      1.518      0.129      -0.012       0.093\n",
       "UK                    0.0507      0.028      1.786      0.074      -0.005       0.106\n",
       "=====================================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_merged['intercept_country'] = 1\n",
    "\n",
    "\n",
    "log_mod = sm.Logit(df_merged['converted'], df_merged[['intercept_country','US', 'UK']])\n",
    "results = log_mod.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "intercept_country    0.130350\n",
       "US                   1.041647\n",
       "UK                   1.052027\n",
       "dtype: float64"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.exp(results.params)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00010001000100010001"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "landing_page\n",
       "new_page    0.118808\n",
       "old_page    0.120386\n",
       "Name: converted, dtype: float64"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.groupby('landing_page').mean()['converted']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:**\n",
    "\n",
    "The restuls of the test indicate that the country does "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='conclusions'></a>\n",
    "## Conclusions\n",
    "\n",
    "The performance of the old page, although slightly better, was deemed to be statistically insignificant. Therefore, the recommendation is to continue using the old page. \n",
    "\n",
    "### Resources\n",
    "https://www.youtube.com/watch?v=7FTp9JJ5DfE&feature=youtu.be\n",
    "\n",
    "Congratulations on completing the project! \n",
    "\n",
    "### Gather Submission Materials\n",
    "\n",
    "Once you are satisfied with the status of your Notebook, you should save it in a format that will make it easy for others to read. You can use the __File -> Download as -> HTML (.html)__ menu to save your notebook as an .html file. If you are working locally and get an error about \"No module name\", then open a terminal and try installing the missing module using `pip install <module_name>` (don't include the \"<\" or \">\" or any words following a period in the module name).\n",
    "\n",
    "You will submit both your original Notebook and an HTML or PDF copy of the Notebook for review. There is no need for you to include any data files with your submission. If you made reference to other websites, books, and other resources to help you in solving tasks in the project, make sure that you document them. It is recommended that you either add a \"Resources\" section in a Markdown cell at the end of the Notebook report, or you can include a `readme.txt` file documenting your sources.\n",
    "\n",
    "### Submit the Project\n",
    "\n",
    "When you're ready, click on the \"Submit Project\" button to go to the project submission page. You can submit your files as a .zip archive or you can link to a GitHub repository containing your project files. If you go with GitHub, note that your submission will be a snapshot of the linked repository at time of submission. It is recommended that you keep each project in a separate repository to avoid any potential confusion: if a reviewer gets multiple folders representing multiple projects, there might be confusion regarding what project is to be evaluated.\n",
    "\n",
    "It can take us up to a week to grade the project, but in most cases it is much faster. You will get an email once your submission has been reviewed. If you are having any problems submitting your project or wish to check on the status of your submission, please email us at dataanalyst-project@udacity.com. In the meantime, you should feel free to continue on with your learning journey by continuing on to the next module in the program."
   ]
  }
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